users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`u-\ w s:z [ji 7neu-\ wou }...

41

Upload: others

Post on 11-Jul-2020

2 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

������������ �������� �������

����� ���! �"��#�$� ���&%('*),+.-/�0"1%2��3432) 5"�� �!6870��"��09:9;�<� �<=0%>�@?BA:=!�#� 6�C<)&=EDGFHA!�

)&=;)&"1IH%JDGFHA$�K��= L:D��0"��NMOD��

P,Q�R�S�T�UHVWTYXGT[Z�T[X�U\Q�T$V^]_QG]a`cbEd2efQgJ��hjia��������,��hk���mlnporq[s��ttu

vcwyx{z�|Jw}c~�z�x��y������~��a~���x����yx��B��������w���x�x{�������m�a���G�mz���xc����w��yx���z�|J~1�������W���W�j~��a�������[~����a�y�{�_����z���w���z�����zx{�����y�������J�����G�����m�J�¡  ���m�{�a�y�p����z��1�p¢�z��y�{�a�yx�£��¤z*����z��J�a¥�����|(�§¦¨�yz��§©���~�z�xJ��~�z�xJ���m�awy�a��x{x{~���x(����z�x�����{�a����z������y�a�y|J���m�2ª«v���¬�|J~����,­��§�B�a�®©¤¯u��°±�a�§�³²´��|J�yx2�j~��a�����y�2�����a~«�¤~����a���{�_��£�}�~�z�xW���aw�x{������a~���x¤w�¢´�����y|J���m�f�mz������mz���x¤|J~1��������x[�aµ�w�~��a���1z���x��~�z������x.���B�a�±�B�a��~���xf�\  z����u~�z����¶���yz��2�yx{��������x�©��z���xu��~�z�xcwy��z�|Jwy�a~���x����y�{�_��������x¶����xu�y����wy�a���������yx¶�mz��·������z�x¨�{�a���m�¸�·  ���m�awy�a��x{xa���1�axu���a������������xc����~����~�¢����mz���x�£

Page 2: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

Page 3: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� ������� ��� � �� ���������

������������ "!#�%$&��� '( )+*�, "�-��.0/+�� "� ,21 .43"$65���.7�8!9. (:(; <=*4>"? �@�8�BAC. * � , (EDF ) �E��G * .4�IHJ� , ��.4��� * ;EF

�LKNM OQPSR&TVU-WYX7Z�[IU-\^]`_badc#ZeKJKfKJKfKJKgKfKJKfKJKfKfKJKgKJKfKJKfKfKJKfKJKgK ����LK�� hCiNaV_jh+_bPNaVU�TdTkZlKJKfKfKJKfKJKfKJKgKfKJKfKJKfKfKJKgKJKfKJKfKfKJKfKJKgK �#m�LK�� ]`U�noTV\Qp-qVr:Z�sot9U-\uKfKfKJKfKJKfKJKgKfKJKfKJKfKfKJKgKJKfKJKfKfKJKfKJKgK �#v

w x 8. ,&y 8.45 ? ���8z , .75 F�({ KNM |QsEU~}�_b}onoP�Z�adi�_bs�KfKfKJKfKJKfKJKgKfKJKfKJKfKfKJKgKJKfKJKfKfKJKfKJKgK �LM{ K�� �~U-\�}�_b}onoP�Z�adi�_bsE\^U�s�iNs7aVU�TkZbq�adi�_bs KJKfKJKfKfKJKgKJKfKJKfKfKJKfKJKgK � {

{ K��LKNM ��_�U���i�\daVU�sEq-UgWoUfWoU�nL��U-\V}��-q-U-\�KJKfKfKJKgKJKfKJKfKfKJKfKJKgK � {{ K��LK�� |QsEUfU-\d}��-q-UJWop��9_bTVUJP���Z�noadTVU�KfKJKfKfKJKgKJKfKJKfKfKJKfKJKgK � {{ K��LK�� ��TdiNadi��7nEU�KfKfKJKfKJKfKJKgKfKJKfKJKfKfKJKgKJKfKJKfKfKJKfKJKgK �9�

{ K�� ]`U-\QU���}�p�Tdi�U�sEq-U-\�KfKfKJKfKJKfKJKgKfKJKfKJKfKfKJKgKJKfKJKfKfKJKfKJKgK �#v� �@�8! , "5%$&��� w@���$&� , $6�@�:� * /���$&. w`(

Page 4: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

{ ����������� �����������������

Page 5: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

��� ���� ����� ���� �

]`U-\ W�s:Z�[ji��7nEU-\ WoU�}�_b}onoP�Z�adi�_bs=aVU�sEWoU�s7a��Jp�sE_bsEq-U�T�WoU-\+P�_bi�\�Wop-q�TdiN�bZ�s7a P�� p��9_� PNnoadi�_bs W�� nosEU _bn�WoU }oPNnE\di�U�noTV\ }�_b}onoP�Z�adi�_bsE\�q-_�U���i�\dakZ�s7a WEZ�sE\ nos�[���[IU=[jiNPNi�U�n`K��U-\ WoU�Tdsoi���TVU-\+\VU Wo_biN�9U�s4a W�����adTVU Z�noakZ�s7a t9p�sEp�TkZ�P�U-\ �7nEU�}�_#\V\Vi��oP�U�U�a��#Z�P�Z��oP�U-\ Z�nE\V\di�oi�U�s Z�n��d_bnoTVW�� r4noio�4n`��Z�n�� _�9U�s 2O^t9U�_bn Z�n aVU�[j}E\8WoU-\�WLiNsE_#\kZ�noTVU-\���U�PNP�U-\`adTV_bno�9U�s7aWo_bsEqfnos iNs4aVp�T���aCiNsEWop�soi�Z��oP�U U�s roi�\daV_biNTVU#K�� PNP�U-\Q\k_bs4a~p�t4Z�P�U�[IU�s4a~adTV�-\^noadiNP�U-\CWEZ�sE\W���Z�noadTVU-\CWo_b[ Z�iNsEU-\���q-_b[j[IU P�� p-q-_bsE_b[ji�U#K�]`U�noTCiN[j}�_bTdakZ�sEq-U U�s��oi�_bP�_bt#i�U�U-\daCp��7i� WoU�[j[IU�s4a � P�Z!�:Zb\VU~WoUQP�U�noT�Wop��9U�P�_b}o}�U�[IU�s4a��:[ Z�i�\�q-U�PNnoi� 2q�i�\%� U-\da�U#"�U-q�adnEpC}oPNnoa�$baakZ�TVWLiN�9U�[IU�s4a�K%� sYU#"�U�a���iNP�ZgR�Z�PNPNn�Z�adaVU�sEWLTVU~P�UCWop&�onoa�WLn�'('CUC\di��-q�P�UC}�_bnoT �9_biNT \VUTVp�Z�PNi�\VU�T^WoU-\^Z��bZ�sEq-p-U-\Q\diNt#soi�):q�Z�adiN�9U-\�K

�JZ�sE\ \V_bs+* ,�,�-/.0,�132547698%2:.<;3=>.?8@6BAC6D8%E08%134F-G:.?E0;��LP�� p-q-_bsE_b[ji�\daVU~Z�sot#P�Z�i�\� Z�PNadr7nE\IH M%�bm#m��M�v�� {7J }oTVp-\VU�s4aVU�U�s M�v����InosEU adrE�-\VUg\kU�P�_bs P�Zb�7nEU�PNP�UgP�Z aVU�sEWEZ�sEq-Uq-_b[j[gnosEUK� aV_bnoaVU-\�P�U-\�U-\d}��-q-U-\ ��iN�bZ�s7aVU-\L� }oTV_bPNiSR p�TVU�T�iNsEWop#)EsoiN[IU�s7a��@Z�n 2WoU�PM��WoU-\[I_�9U�sE\�WoUC\dnN�E\di�\dakZ�sEq-U~WLi�\d}�_bsoi��oP�U-\���U-\da�P�Zg}oTdiNsEq�iN}:Z�P�UJq�Z�nE\VUCWoUCP���Z�TdT���a WoU-\�}oTV_� t#TV�-\=WoU P�� r4no[ Z�soiNaVp#KLOLZ�sE\=R&TVU�iNs3��P�Z q�TV_bi�\k\kZ�sEq-U W�� nosEU }�_b}onoP�Z�adi�_bs U-\Va t9p-_b[Ip# adTdi��4nEUF��akZ�sEWLi�\j�7nEU=q-U�PNP�U�WoU-\�}oTV_�WLnEq�adi�_bsE\jWoU=P�Z aVU�TdTVU�U-\dajZ�n [ji�U�nL� Z�TdiNadro[Ip# adi��4nEU#K8��U-\g[I_�9U�sE\K)Esoi�\V\VU�s7a Wo_bsEqI}:Z�T!��adTVUIiNsE\dnP \kZ�s7aV\RQ q-U�R&noagP�Uj}�_biNs7a WoUWop�}:Z�Tda~W�� nosEU }oroiNP�_#\V_b}oroi�UF��P�U [ Z�PNadr4nE\di�Z�soi�\V[IUF��WEZ�sE\QP�Zb�7nEU�PNP�U P�U q-_bs7adT�$bP�UjWoUgP�Zs:Z�akZ�PNiNaVpJ}:Z�T P�Z�q-_bs4adTkZ�iNs7aVUg[I_bTkZ�P�Ugadi�U�s4a nosEUft#TkZ�sEWoUJ}oP�Zbq-U�SNM#T2K

O^}oTV�-\~q-U�P�Z�@P�U [ Z�adrEp�[ Z�adi�q�i�U�sU��U�PNt9U O�Wo_bPN}orEUWVQnEp�aVU�P�U�a�H M%��X#m��M�v9� {7J p�[jiNaP�� rY�}�_badrE�-\VU�\VnoiN�bZ�s7aVU[Z�P�U-\ }�_b}onoP�Z�adi�_bsE\ \V_bs4a \VakZ��oiNPNi�\Vp-U-\\�QP�ZQR _bi�\�}:Z�T P�U�noT }�_baVU�s adi�U�P+WoUjq�TV_bi�\V\kZ�sEq-Ujt9p-_b[Ip�adTdi��7nEUj[ Z�i�\fp�t4Z�P�U�[IU�s7ag}:Z�TJnos R�Zbq�aVU�noTgWoU�TVp-\di�\dakZ�sEq-UZ�n Wop��9U�P�_b}o}�U�[IU�s4a+}oTV_b}�_bTdadi�_bsosEU�P:Z�njq�Z�TdTVp WLn akZ�nL��WoU�q�TV_bi�\V\ Z�sEq-U#K�] � p-qVrE_Jt#TkZ�s WLi�\V\kZ�s4a Wo_bsosEp]�fP�ZfadrEp-_bTdi�UCWoU(� Z�PNadr4nE\ U�a�P�� nE\ Z�t9U^aV_bn��d_bnoTV\�}oPNnE\ iN[j}�_bTdakZ�s4a�WoU-\\dakZ�adi�\dadi��7nEU-\gU�s \Vq�i�U�sEq-U-\gr7no[ Z�iNsEU-\ [IU�sE��TVU�s4a�nos Z�noadTVU=[ Z�adrEp�[ Z�adi�q�i�U�s^��U�PNt9UF�_ i�U�TdTVU# a`oTkZ�s%b-_bi�\Jh+U�Tdr7noP�\da�H M�v�� { ��M�v { X J �@nos p�adnEWLi�Z�s4afWoUcVQnEp�aVU�P�U�a��>� P���Z�}o}oPNi�q�Zd adi�_bs WoUfP�ZjadrEp-_bTdi�U WoU-\^}oTV_��:Z��oiNPNiNaVp-\]�IP�� p-q-_bsE_b[ji�Ug}�_bPNiNadi��7nEUgU�a��E}oPNnE\Q}:Z�Tdadi�q�noPNi��# TVU�[IU�s4a���Z�nL��p�adnEWoU-\ Wop�[I_bt#TkZ�}oroi��7nEU-\�K�e P�Wop��9U�P�_b}o}:Z P�� p-�4n:Z�adi�_bs �7noi }�_bTdaVU \V_bssE_b[fZ gihgij+kml h�n<jporqUs3hutin<j�o_�v h TVU�}oTVp-\kU�s4aVU P�U^sE_b[K�oTVUQW�� iNsEWLiN�7i�WLnE\�W�� nosEU }�_b}onoP�Z�adi�_bs`KY� PNP�UQU-\da+}oPNnE\�q-_bsos4nEU\V_bnE\�P�U~sE_b[ W��Mwpxzy{F|~}����U������}?�#|~}�x&y%�WSNMzT2K

]+� p�akZ�}�UC\VnoiN�bZ�s7aVUCWEZ�sE\�P�UCWop��9U�P�_b}o}�U�[IU�s7a WoU~q-UQWo_b[ Z�iNsEU~WoU-\��oi�_b[ Z�adrEp�[ Zd adi��4nEU-\f�7nEUI\V_bs7afP�U-\fW�s:Z�[ji��4nEU-\ WoUj}�_b}onoP�Z�adi�_bs ZYp�aVp�R TkZ�sEqVroi�UI}:Z�TfP�Uj[ Z�adrEp#

Page 6: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

m ��� ����� ���� ������

[ Z�adi�q�i�U�sYiNakZ�PNi�U�s�hCiNaV_gh+_bPNaVU�TdTkZ�H M�v#m����M�X { � J �L�4noi ��\dnoT�P�Z[�:Zb\VUCWoU-\ _��E\VU�Td�#Z�adi�_bsE\WoU \V_bsB��U�Z�n �)EP�\Q|Q[K��U�TdaV_ � � OQsEq-_bs:Z H M�vFX#m�"M�X#m {7J ��ZIp�akZ��oPNi�P�U-\Qp-�7n:Z�adi�_bsE\Q�7noiTVp�t#i�\V\kU�s4a P�Zjq-_br:Z��oiNakZ�adi�_bs _bn�P�Z�q-_b[j}�p�adiNadi�_bs U�s4adTVUgWoU�nL�YU-\V}��-q-U-\[Zgihgij k l h0n<j�orq�s3h�n<jpo n<jpo q �

h t n<jpoggYj k �

n<j�orq h0n<j�o

n<j�o�q

� t n<j�o

_�v h U�a TVU�}oTVp-\VU�s7aVU�s4a�P�U^sE_b[K�oTVU~W�� iNsEWLiN�7i�WLnE\ WoUCqVr:Zb�7nEUQU-\V}��-q-U#K � PNP�U-\�}�_bTdaVU�s7aP�U sE_b[ W��Mwpxzy{F|~}����N���7��C�F|���{����N����|a�����p{F�9OQPSR TVU-W X7Z�[IU-\+]`_badc#ZcH M�v#v���EM�X { X J Zz4Z�s7at#TkZ�sEWoU�[IU�s7aCq-_bPNP�Z���_bTVpK� P�U�noT^Wop��9U�P�_b}o}�U�[IU�s4aISNM#T2K

]8ZfadrEp-_bTdi�UCWoU-\�W�s:Z�[ji��7nEU-\�WoU^}�_b}onoP�Z�adi�_bs�ZfTVp-U�PNP�U�[IU�s4a �7n=P�Ur�d_bnoT �f}:Z�TdadiNTWoU-\�Z�sosEp-U-\QM�X�����7[ Z�i�\ �~P�� p�}�_��4nEUF�4P�U-\ \VU�noP�\ �bZ�Tdi�Z��oP�U-\�}oTdi�\VU-\ U�s q-_b[j}oaVU^p�akZ�i�U�s7aP�UJsE_b[!�oTVU W�� iNsEWLiN��i�WLnE\CWoU-\ }�_b}onoP�Z�adi�_bsE\^q-_bsE\di�Wop�TVp-U-\��:P�� iNs��EnEU�sEq-U WoUfP�� p��9_bPNnoadi�_bst9p�sEp�adi��7nEU0p�akZ�s7a sEp�t#PNiNt9p-U#K��C� U-\Va U�s R Z�iNa P�_bTV\V�4nEU�P�U-\I\Vq�i�U�s4adi�):�7nEU-\ U�s �oi�_bP�_bt#i�UZ�soiN[ Z�P�U �9_bnoPNnoTVU�s4a�\VUjWLi�\dadiNsot#nEU�T WoUIP�Z��oi�_bP�_bt#i�Uj�9p�t9p�akZ�P�U �7nEUjP�U-\fW�s:Z�[ji��7nEU-\WoU }�_b}onoP�Z�adi�_bs q-_bsos4noTVU�s7a nosjTVp-U�P:U-\k\V_bT��bq�Z�T P�� p�adnEWoU W�� nosEU }�_b}onoP�Z�adi�_bs=\VU adTV_bno�9U��adTVUJP�U�� nE\daVU~[jiNPNi�U�n U�s4adTVUfq-U�PNP�UfW�� nos _bTdt4Z�soi�\d[IUfU�a^q-U�PNP�UfW�� nosEUgq-_b[j[gnos:Z�noaVp#K

]"U-\I}oTVU�[ji�U�TV\IadTkZ��bZ�nL� \dnoTjP�Z }�_#\V\di��oiNPNiNaVpYW�� nosEU�TVU�}oTVp-\VU�s4akZ�adi�_bs [ Z�adrEp�[ Zd adi��7nEU WoU-\�W�s:Z�[ji��4nEU-\ WoUe}�_b}onoP�Z�adi�_bs _bs4a p�aVp�Zbq-q�nEU�iNPNPNi�\ Z��9U-q5��U�Z�nEq-_bno} WoU\Vq-U�}oadi�q�i�\V[IU#Kue P�\�R noTVU�s4aQsEp�Z�so[I_biNsE\QP�Ug}�_biNs7aQWoU Wop�}:Z�TdaCWoU-\ �oi�_b[ Z�adrEp�[ Z�adi��4nEU-\%KOQ}oTV�-\^P�Z=�~U�nL��i���[IU��CnEU�TdTVU!� _bsEWLi�Z�P�UF��P�U-\CW7s:Z�[ji��7nEU-\QWoU }�_b}onoP�Z�adi�_bs p�akZ�i�U�s7aTVU�P�Z�adiN�9U�[IU�s7aR�oi�U�s Wop��9U�P�_b}o}�p-U-\���[ Z�i�\CU�PNP�U-\Cp�akZ�i�U�s7afWLiN�7i�\kp-U-\QU�s adTV_bi�\~Wo_b[ Z�iNsEU-\WLi�\dadiNsEq�aV\ Zb\V\VU��YrEU�Td[Ip�adi��7nEU-\ P�U-\InosE\=Z�nL� Z�noadTVU-\BZ P�U-\jTVU-qkrEU�TVqVrEU-\ adrEp-_bTdi��7nEU-\��q-U�PNP�U-\�U#"�U-q�adnEp-U-\�U�s P�Z���_bTkZ�aV_biNTVU-\�U�a�U�s)Es q-U�PNP�U-\Y�4nEU P�� _bs R�Z�i�\kZ�iNa�\dnoT�P�U aVU�T TkZ�iNs`K#]`U-\+TVU-qVrEU�TVqVrEU-\ adrEp-_bTdi��4nEU-\ p�akZ�s4a+[IU�sEp-U-\�}:Z�T+WoU-\�[ Z�adrEp�[ Z�adi�q�i�U�sE\�s`��Zz4Z�s7a�7n`� nosY\VU�sE\ PNiN[jiNaVpJWoUCP�ZK�oi�_bP�_bt#i�UF�LP�U-\ TVp-\VnoPNakZ�aV\�_��oaVU�s4nE\ sEUQR noTVU�s4a�}:Zb\�Z�i�\Vp�[IU�s7aZ�}o}oPNi�q�Z��oP�U-\�U�a�P�U-\\�oi�_bP�_bt#i�\VaVU-\�\kU adTV_bno�9��TVU�s4a \kq�iNsEWop-\�U�s�WoU�nL� t#TV_bno}�U-\��4q-U�nL�=�7noiP�U-\^Zbq-q-U�}oaV��TVU�s7aCU�a q-U�nL���7noi�P�U-\�TVU�R nE\V��TVU�s4a�Ke P9p�akZ�iNa�Z�P�_bTV\"q�P�Z�iNT"�4n`� iNP4s`� ~Z�noTkZ�iNa8}:Zb\`WoU P�_bi4\diN[j}oP�U U�a`�4nEU P�U Wop��9U�P�_b}o}�U�[IU�s4aWoU aVU-qVrosoi��7nEU-\IW���Z�s:Z�P�L\VU�}oPNnE\j\V_b}oroi�\dadi��7nEp-U-\jp�akZ�iNa�sEp-q-U-\V\kZ�iNTVU#K ]`U-\I�oi�_bP�_bt#i�\VaVU-\s`� p�akZ�i�U�s7aj}:Zb\ adTV�-\c�0P���Z�i�\VU WEZ�sE\ P�U-\�Wo_b[ Z�iNsEU-\�}onoTVU�[IU�s7ajadrEp-_bTdi��4nEU-\���aV_bnE\ sEU}:Z�TdakZ�t9U�Z�i�U�s7aj[���[IU }:Zb\ P�U [���[IU=}�_biNs4a�WoUI��nEU=�7n:Z�s4ajZ�nL�eR Z�iNaV\ U�nL� �[���[IU-\�K�C� U-\daJ}�_bnoTV�7nE_bi �`Wo�-\IM�X � ���iNP�\JadTkZ��bZ�iNPNP���TVU�s7aI��\VU!)o�oU�TJnos�_���dU-q�adiSR q-_b[j[ nos�U�a�I[ji�U�nL�Yq-_bsEq�iNPNi�U�TCP�U-\ TVp-\VnoPNakZ�aV\^adrEp-_bTdi��7nEU-\CZ��9U-qfP�U�noTV\C_��E\VU�Td�#Z�adi�_bsE\�K:]`Ug[I_bsEWoUTVp-U�P"U�a \k_bs0U���}oP�_bTkZ�adi�_bs�R&noTVU�s7aQZ�P�_bTV\ }oP�Zbq-p-\CZ�n�q-U�s4adTVU#K

O }:Z�TdadiNT WoUjM�X#m���LP���Zbq-q-U�s4a�R noa [ji�\ \dnoT�WoU-\ }oTV_��dU�aV\ WoUCt#TkZ�sEWoUfp-qVrEU�PNP�UfqkrEU�T qkr:Z�s4aK� Z�s:Z�P��\VU�TJWoU-\~p-q-_#\pL\daV��[IU-\~U�s4adi�U�TV\�K�]8Z��oi�_bt9p-_�qVroiN[ji�UIU�aCP�U-\~p�adnEWoU-\J\dnoTP�U-\��EnL��W�� p�sEU�Tdt#i�U^U�a+\dnoT�P�U-\ TkZbWLi�Z�adi�_bsE\3� _bnE��TVU�s7a+nosjT�$bP�U�WoU }oTVU�[ji�U�T _bTVWLTVU#K! 8_bnoaq-U�P�Z=R noafZbq-q-_b[j}:Z�t#sEpIWoU�P�� noadiNPNi�\kZ�adi�_bs�WoU sE_bno�9U�Z�nL� iNsE\dadTdno[IU�s4aV\��"WoU�sE_bno�9U�PNP�U-\aVU-qkrosoi��4nEU-\QU�a W�� nosEU~TdiNt#nEU�noT sE_bno�9U�PNP�UF�:Z�nE\V\di`WEZ�sE\ P�� nE\ Z�t9UfWoU-\ [ Z�adrEp�[ Z�adi��4nEU-\%K

Page 7: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� � ��� � ��� ��� �� �

�C� U-\da\�Jq-U�adaVU p�}�_7�7nEU��7nEU P�Z]�oi�_bP�_bt#i�U���iNa+nosEU WoU�q-U-\r�oTkZ�sEqkrEUQq-_bsos:Z � adTVU�nos iNs4aVp# T���a8t#TkZ�sEWLi�\V\ Z�s4a��F�^\ Z-�9_biNT�P�ZL�oi�_bP�_bt#i�U �7n:Z�s4adiNakZ�adiN�9U�U�a�\dakZ�adi�\dadi��7nEUF� �7noi�\kU+aV_bnoTds:Z�iNa�9U�TV\+WoU-\ Wo_b[ Z�iNsEU-\��4nEU�P��7nEU�}�U�n p-\V_baVp�Tdi��7nEU-\ q-_b[j[IU�P�Z~adrEp-_bTdi�UQWoU-\+\pL\daV��[IU-\��bP�ZadrEp-_bTdi�UjWoU P�� iNsLR _bTd[ Z�adi�_bs3��P�Z }oTV_bt#TkZ�[j[ Z�adi�_bs [ Z�adrEp�[ Z�adi��7nEUIU�aJP�Z=adrEp-_bTdi�UjWoU-\� U�nL�US���Ne2s7adTV_7WLnEq�adi�_bs3�:}o}O����dT2K

]+� iN[j}�_bTdakZ�sEq-UjWLn Wop��9U�P�_b}o}�U�[IU�s7agWoU [I_7Wo��P�U-\~[ Z�adrEp�[ Z�adi��7nEU-\JTVp�Z�PNi�\daVU-\JU�a}oTVp-q�i�\ Z R�Z-�9_bTdi�\VpIP�� noadiNPNi�\kZ�adi�_bs WoU-\f[Ip�adrE_7WoU-\ \dakZ�adi�\dadi��7nEU-\fU�a Z�TVU�sEWLn�P�U-\g_bTVWLi� s:Z�aVU�noTV\JiNsEWLi�\d}�U�sE\kZ��oP�U-\�K _ Z�TkZ�PNP���P�U�[IU�s4a���nos�Z�noadTVUIq-_bnoTkZ�s7agadrEp-_bTdi��7nEUI\�� U-\dafWop# �9U�P�_b}o}�p=Wo_bs4agP�UW�onoags`� p�akZ�iNa }:Zb\gWoU-\ }oTVp��7i�\Vi�_bsE\fs4no[Ip�Tdi��7nEU-\��8[ Z�i�\ \kU�noP�U�[IU�s4aWoU-\ TVp-\dnoPNakZ�aV\ �7n:Z�PNiNakZ�adiSR \�KE�C� U-\da \VnoTdaV_bnoa \V_bnE\ P�� iNs��EnEU�sEq-UJWoU��^_���U�Tda�� K� Zbq-O^T adr4noTWH M�X����� M�X9�G� J �7nEU�q-Ujq-_bnoTkZ�s4a Z=��n P�U(�d_bnoT�K:e P�}�U�sE\kZ�iNaJ�7nEU }�_bnoTfp-q�P�Z�iNTVq�iNTnose}oTV_��oP���[IU \di �#Zb\daVUF��iNP+p�akZ�iNafiNsEWLi�\V}�U�sE\kZ��oP�U WoUjP�UjTVp-\V_bnEWLTVUW��\ Z }oPNnE\g\diN[j}oP�UU���}oTVU-\V\Vi�_bs`K |^sEUY�7nEU-\dadi�_bsB\VU�}�_#\kZeZ�P�_bTV\9Z�� nE\V�7n`� _�v }�U�noa 2_bs \�� p�P�_biNt#sEU�T=WoU�P�ZTVp�Z�PNiNaVpf\kZ�sE\�aV_bakZ�P�U�[IU�s7a^P�Z�}�U�TVWLTVUfWoU~��nEU��e2P�\���Z�t#i�\V\kZ�iNa�U�s R Z�iNa�WoU�\kZ��9_biNT \di4P�Z^TVU-qVrEU�TVqkrEU�WoU��bZ�iNa Z-�9_biNT WoU-\0�onoaV\"}onoTVU�[IU�s7a}oTkZ�adi��4nEU-\+_bnI\ViEU�PNP�U WoU��bZ�iNa+\kU�Td�7iNT\�~WoU-\�)EsE\+}oPNnE\ Z��E\dadTkZ�iNaVU-\�K9]8Z~TkZ�i�\V_bsIW�����adTVU WoUP�Z��oi�_bP�_bt#i�UgU-\da�nosEU~[IU�iNPNP�U�noTVU q-_b[j}oTVp�rEU�sE\di�_bs WoU-\^TVU�P�Z�adi�_bsE\^U�s4adTVUfP�U-\ WLi�"�p�TVU�s7aV\}oTV_bakZ�t9_bsoi�\daVU-\JWoU P�Z s:Z�adnoTVU#K 8_bnoafZ�n P�_bsotYWoU�\V_bs roi�\daV_biNTVUF��q-U�adaVUj\Vq�i�U�sEq-U�\�� U-\da�:Z�adadnEUfq-_bs4adTVUgP�� i�Wop-UfW�� nosEUJs:Z�adnoTVUf_�v�aV_bnoa U-\Va^qVr:Zb_badi��7nEUfU�a^q-_bs7adiNsot9U�s4a�K

]"Z �oi�_bP�_bt#i�U�}oPNnE\����4nEU P�U-\JZ�noadTVU-\J\Vq�i�U�sEq-U-\���adTkZ�iNakZ�iNaR��U�Z�nEq-_bno} WoU �7nEU-\dadi�_bsE\W�� nosEU~[ Z�soi���TVU~roi�\daV_bTdi��4nEU#Ko]"U-\�adrEp-_bTdi�U-\ p�akZ�i�U�s4a�adiNTVp-U-\^W�� _��E\VU�Td�bZ�adi�_bsE\�R Z�iNaVU-\ Z�n}oTVp�Z�P�Z��oP�U#K`]"Z [I_�Wop�PNi�\kZ�adi�_bs [ Z�adrEp�[ Z�adi��7nEU U�s W�s:Z�[ji��4nEU-\fWoU�}�_b}onoP�Z�adi�_bs sEU\dnoiN�bZ�iNa~}:Zb\fq-U�adaVUI[Ip�adrE_7WoU#K�|Qs \Vq-p�s:Z�Tdi�_Yq-U�TdaVU-\f\diN[j}oPNi�):pI[ Z�i�\J}oP�Z�nE\di��oP�UIp�akZ�iNa}oTVp-\VU�s4aVpQU�a�q-U�s`� U-\da��4n`��Z�}oTV�-\��4nEU P�U-\ R�Z�iNaV\ p�akZ�i�U�s4a _��E\VU�Td�9p-\�}�_bnoT�q-_bs)ETd[IU�T�_bn3�P�U+q�Zb\8p-qkrEp�Z�s4a�� iNs)ETd[IU�T�q-U�WoU�Tdsoi�U�T�K�� ��[IU \di7q-U-\"WoU�nL�f[Ip�adrE_�WoU-\8W�� iNs7�9U-\dadiNt4Z�adi�_bs_bs4a }�_bnoTL�onoa^q-_b[j[ nos WoUJ[ji�U�nL��q-_b[j}oTVU�sEWLTVUgP�UJ[I_bsEWoUg�7noi�sE_bnE\^U�s7aV_bnoTVUF�EP�U-\WLi�"�p�TVU�sEq-U-\C�4noi8P�U-\^\Vp�}:Z�TVU�s7aC_bs7aCp�aVpgP�Zj\V_bnoTVq-U WoUR�oi�U�s WoU-\^q-_bs4adTV_%�9U�TV\VU-\�K`�C� U-\dat#TkZbq-U �Jq-U�adaVUQ�4nEU-\dadi�_bs=U�a��~aV_bnE\ P�U-\+adrE��[IU-\�WLiN�7i�\kZ�s7a P�U^[I_bsEWoU^\Vq�i�U�s7adi�):�4nEUQ�4nEUP�Z�q-_7U��Li�\daVU�sEq-UfWoUJ}oPNnE\Vi�U�noTV\^p-q-_bP�U-\^WoU]�oi�_bP�_bt#i�U Zjp�aVpJ}�_#\V\di��oP�UcS���%e s4adTV_�WLnEq�adi�_bs3�}o} { � � T2K

O^n aV_bnoTds:Z�s7a WLn \di��-q�P�UF� WoU t#TkZ�sEWoU-\ Z-�#Z�sEq-p-U-\0_bs4a p�aVp R Z�iNaVU-\�}:Z�T0]"_badcbZ�nosEU[)Et#noTVU }:Z�Tdadi�q�noPNi���TVUjWEZ�sE\CP�� roi�\daV_biNTVU�WoU P�Z \Vq�i�U�sEq-UF��}oTVU-\V�7nEU [ Z�PNt#TVp PNnoi�K3O�_bsi�Wop-UjWoU�P�Z s:Z�adnoTVUIU�seakZ�s4af�7n`� nosoiNaVpIWo_bs4agP�� rE_b[j[IU�R Z�iNaJ}:Z�Tdadi�U�adTkZ�sE\V}:Z�TkZ�i�\V\kZ�iNaWEZ�sE\^\VU-\^adTkZ-�#Z�nL��}:Z�TQ\kZ�aVU�sEWEZ�sEq-U��IqkrEU�TVqVrEU�T~WoU-\^P�_bi�\^t9_bno�9U�Tds:Z�s4aJq-U�adaVUfnosoiNaVpU�a��`U�s4adTVU�Z�noadTVU-\���nos�iNs4aVp�T���ag}oTV_bsE_bsEq-p=}�_bnoTgWoU-\g�7nEU-\dadi�_bsE\ aVU�PNP�U-\g�7nEUIq-U�PNP�U WoUP�� _bTdiNt#iNsEUQWoUQP�Zgq-_bsE\kq�i�U�sEq-U^r7no[ Z�iNsEU#KoO q-U�adaVUQp�}�_7�7nEUF�9P�ZK�oi�_bP�_bt#i�UCU�a�P�U-\�\Vq�i�U�sEq-U-\\V_7q�i�Z�P�U-\Qp�akZ�i�U�s4aJU�s0}oP�U�iNs U-\V\V_bTCU�a^P�U-\ �oi�_bP�_bt#i�\daVU-\C_7U�no��TkZ�i�U�s4a[�jP�Z q-_bsE\dadTdnEq�adi�_bs

Page 8: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

v ��� ����� ���� ������

W�� nosEU WLi�\Vq�iN}oPNiNsEU R Z�i�\kZ�s7a P�Z0\p�s4adrE�-\VU WoUIP�� p�adnEWoU�WoUIP�� U-\V}��-q-U r4no[ Z�iNsEU�U�s�akZ�s7a�7nEU~}:Z�Tdadi�UfiNs7aVp�t#TkZ�s4aVUgWoUfP�Z s:Z�adnoTVUWS���Ne s4adTV_�WLnEq�adi�_bs3�E}`K:mdT2K

hCiNaV_�h+_bPNaVU�TdTkZ Z+� _bnEp nos T�$bP�U WoU }oTVU�[ji�U�TY_bTVWLTVU WEZ�sE\ P�U0Wop��9U�P�_b}o}�U�[IU�s4aWoU P�Z+�oi�_bP�_bt#i�U=adrEp-_bTdi��4nEU#K0e2P�}�_bTdakZ�iNa nos�iNs7aVp�T���a }:Z�Tdadi�q�noPNi�U�TjWEZ�sE\ P���Z�}o}oPNi�q�Z�adi�_bsWoU-\Q[ Z�adrEp�[ Z�adi��4nEU-\JZ�nL�0\Vq�i�U�sEq-U-\QsE_bs0}or7�\Vi��4nEU-\��@U�s0}:Z�Tdadi�q�noPNi�U�T~P�Zc�oi�_bP�_bt#i�U U�aP�� p-q-_bsE_b[ji�U#KL� ��[IU \�� iNP~s`��Z���TkZ�iN[IU�s4a�q-_b[j[IU�sEq-p5��adTkZ-�#Z�iNPNP�U�T�\dnoT�P�Z �oi�_bP�_bt#i�U�7nEU��9U�TV\g\V_biS�EZ�s4aVU=Z�sE\���iNP R noaJadTV�-\J}oTV_7WLnEq�adiSR U�aJ}onN�oPNi�Z9��U�Z�nEq-_bno}�WoU�TVp-\VnoPNakZ�aV\� nE\V�7n`� � \kZ [I_bTda���[���[IU�\di8P�Z=[I_bs7aVp-UjU�s }onoi�\V\ Z�sEq-U WLn R�Zb\Vq�i�\d[IU U�s e akZ�PNi�UF��Wo�-\M�X�����oTVU�sEWLiNaQ\V_bs�adTkZ-�#Z�iNP"WoU~}oPNnE\QU�sY}oPNnE\^WLi�P q�iNP�U�S��dT2K

�CZz�[I_bsEW _ U�Z�TdP/H M�v9��X�oM�X { � J �#Wo_bs4a _bsIWLiNa+�7n`� iNP:ZCWop-q-_bno�9U�Tda ]`_badc#Z�9Z-�#Z�iNa+P�U[���[IU~}�_biNs4a WoUC��nEUC�7nEU~q-U~WoU�Tdsoi�U�T�KEO P�� iNsE\dakZ�T WoU~\V_bsYp�P����9UF�LiNP�s`� p�akZ�iNa�}:Zb\ �oi�_� P�_bt#i�\daVU#KO�_bs adTkZ��bZ�iNP�\�� iNsE\Vq�TdiNa�WEZ�sE\ P�Z~aVU�sEWEZ�sEq-UQt9p�sEp�TkZ�P�UQU�s p-q-_bP�_bt#i�U]�~P�� p�}�_7�7nEU#Ke P }�_bTdakZ�iNagnos TVU�t4Z�TVW \diN[jiNP�Z�iNTVUI\VnoTfP�Z�\V_�q�i�p�aVpjr4no[ Z�iNsEU=U�af\dnoTfnosEUIq-_bPNP�U-q�adiN��iNaVpZ�soiN[ Z�PNi���TVU#KE] � roi�\daV_biNTVUgZ�noTkZ�\dnoTdaV_bnoa�TVU�aVU�s7n0WoUCPNnoi�P�� U���adTkZb_bTVWLiNs:Z�iNTVUfq-_bs7adTV_��9U�TV\VU\dnE\kq�iNaVp-U }:Z�T P�U [I_7Wo��P�U�P�_bt#i�\Vadi��4nEUF�biNsE\d}oiNTVp }:Z�T P��Mwpx&yN{F|~}~�F� �Y� �%�����Yy � �#|��9�7n`� iNPo}oTVp# \VU�s7akZ�WEZ�sE\ P�U-\�Z�sosEp-U-\fM�X���� q-_b[j[IU~nosEUQP�_bi�WoU-\�W�s:Z�[ji��4nEU-\�WoUC}�_b}onoP�Z�adi�_bs�S���e s4adTV_�WLnEq�adi�_bs3�E}`K@��T2K

8_bnE\�q-U-\grE_b[j[IU-\�WoU \kq�i�U�sEq-U s`��Z�noTkZ�i�U�s7a }:Zb\ U�n�WoU aVU�P�\g}:Z�TVq-_bnoTV\�\kZ�sE\ P�U�onoa��4n`� iNP�\�\�� p�akZ�i�U�s4a\)o�Lp!Z#[IU�adadTVU P�Zfs:Z�adnoTVU^U�s p-�7n:Z�adi�_bs`K _ _bnoT�q-U�P�Z�7qVr:Zbq�nos nE\kZWoU \kZ [Ip�adrE_�WoU#K�h+_bPNaVU�TdTkZ� }:Z�T�U��LU�[j}oP�UF��}�_bnoT Z�}o}oPNi��7nEU�T=P�U-\=[ Z�adrEp�[ Z�adi��4nEU-\U�s7adTVU Z�noadTVU-\��YP�Z��oi�_bP�_bt#i�U=�9_bnoP�Z�iNa ZbWEZ�}oaVU�T P�U-\f[I_�Wo��P�U-\gU�a P�U-\gq-_bsEq-U�}oaV\ WoUIP�Z[Ip-q�Z�soi��7nEU q�P�Zb\k\di��4nEU U�a=q-_bsE\dadTdnoiNTVU nosEU�sE_bno�9U�PNP�U [Ip-q�Z�soi��7nEU �7noi^TVp�t#iNTkZ�iNa P�U-\Zb\V\k_7q�i�Z�adi�_bsE\ �oi�_bP�_bt#i��7nEU-\�S { � _ TVp�R Zbq-U#T2K

]"_badcbZ�4P�_bTV\V�4n`� iNP:}oTVp-\VU�s4aVUQP�U \dn��dU�a WoU \V_bs adTkZ-�#Z�iNP@WEZ�sE\ * 4Y6�� 6@;:G:,KE��������,d.?=u-/4� ].<E/47E���\��}:Z�TdP�UjWoUI�oi�_bP�_bt#i�U�}or7�\Vi��4nEUF��aVU�Td[IU�}:Z�TJP�U-�7nEU�P iNP Wop-\diNt#sEU�P�U-\Z�}o}oPNi�q�Z�adi�_bsE\�WoU-\I}oTdiNsEq�iN}�U-\ U�a WoU-\I[Ip�adrE_�WoU-\ }or7�\Vi��4nEU-\ Z�nL� \��\daV��[IU-\W�oi�_bP�_� t#i��7nEU-\�KNe P`q-_bsE\kZbq�TVUJ[���[IUJnos0qkr:Z�}oiNadTVUgU�s7adi�U�TCZ�nY}oTV_bt#TkZ�[j[IUgWoUgq-U�adaVUR�oi�_bP�_bt#i�U}or7�\di��7nEU#Kb]`U-\�t#TkZ�sEWoU-\�PNiNt#sEU-\�U�s�\k_bs4a�P�U adTkZ�iNaVU�[IU�s7a \pL\daVp�[ Z�adi��4nEUF��W�� _�vjWop-q-_bnoP�UZ�P�_bTV\QnosEUgWLiN��i�\di�_bs0s:Z�adnoTVU�PNP�U�WLn \dn�� U�aQU�s \V_bnE\ 2\kU-q�adi�_bsE\�K@e P"}oTVp-\VU�s7aVUgP�Z��oi�_bP�_bt#i�U}or7�\di��7nEUCq-_b[j[IU^nosEU �oTkZ�sEqVrEU~WoU P�Zf[Ip-q�Z�soi��4nEUQt9p�sEp�TkZ�P�U~WoU P�� p��9_bPNnoadi�_bs`KE��U�PNP�U# q�iEadTkZ�iNaVUQWoU-\ \pL\daV��[IU-\+Wo_bs7a P�� _bTdt4Z�soi�\kZ�adi�_bs�\dnN�oiNa WoU-\ qVr:Z�sot9U�[IU�s4aV\ iNTdTVp��9U�TV\Vi��oP�U-\S � � _ TVp�R Zbq-UfU�aQ��r:Z�}oiNadTVU � Z rEU _ TV_bt#TkZ�[ _�R _ r7�\Vi�q�Z�P���i�_bP�_btFYT2K

] ��Z�s:Z�P��\VUf[ Z�adrEp�[ Z�adi��4nEUF�@U�a Wo_bsEqJP�Z [I_7Wop�PNi�\ Z�adi�_bs3�EWo_biNa \VUJq-_bsEq-U�s4adTVU�TC\dnoTnos sE_b[K�oTVU TVU-\dadTVU�iNs7a WoU Wo_bsosEp-U-\���U�a }:Z�T PM�d �[���[IU=U�PNP�U U-\da U�s q-_bs��EiNajZ-�9U-q P�Z��i�\di�_bs roi�\daV_bTdi��7nEU#K0� Z�i�\�q-U�adaVU�q-_bsLR&TV_bs7akZ�adi�_bsB\VU�TkZ nos [I_baVU�noT��7noiL��}oPNnE\Vi�U�noTV\TVU�}oTdi�\VU-\^}�U�Td[IU�adadTkZ nosEUJ[IU�iNPNP�U�noTVU q-_b[j}oTVp�rEU�sE\di�_bs WoUJP�Z�s:Z�adnoTVU U�a^WLn�T�$bP�Ug�7nEUP�U-\�[I_�Wo��P�U-\ [ Z�adrEp�[ Z�adi��7nEU-\C_bs4a � �d_bnEU�T^U�s+�oi�_bP�_bt#i�U#K

Page 9: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� � ��� � ��� ��� �� X

]"Z [ji�\VU�U�sep-�4n:Z�adi�_bs�WoU P�Z��~Z�adnoTVU�}:Z�TJP�� rE_b[j[IUjU-\daJnosEU�Wop�[ Z�TVqVrEUIR Zb\Vq�i� s:Z�s4aVU#K:�~U~P�ZI�:Zb\Vi��4nEUJt9p-_b[Ip�adTdi�U WoU-\ �CTVU-q-\^}�U�sEWEZ�s7a P�� OQs4adi��7noiNaVp!��P�Z�}or7�\Vi��4nEUZbq�adnEU�PNP�U P�Z=}oPNnE\J\V_b}oroi�\Vadi��4nEp-UF��P�� U-\d}oTdiNa~r4no[ Z�iNs�Z=\dn3��}�U�sEWEZ�s4afWoU-\C[jiNPNP�p�s:Z�iNTVU-\��U���adTkZ�iNTVUJWLnY\d}�U-q�akZbq�P�UJ�4noi�\�� _�"�TVUR� \VU-\ 9U�nL� WoU-\ }oTdiNsEq�iN}�U-\�P�_bt#i��7nEU-\�KNe2P�U-\da aV_bnoaWoUJ[���[IUg\dnoTd}oTVU�s:Z�s7aC�7n`� nosEUfWLi�\Vq�iN}oPNiNsEU Z�noaV_bsE_b[IU q-_b[j[IUgP�U-\ [ Z�adrEp�[ Z�adi��7nEU-\Z�iNa^Z�noakZ�s4aQW���Z�}o}oPNi�q�Z�adi�_bsE\QU�a^�7nEUJq-U�PNP�U-\ 2q�i"\V_bi�U�s7aQZ�noakZ�s4a^WLiN�9U�TV\Vi�):p-U-\KZ�oi�_bP�_bt#i�UF�}orYL\di��4nEUF�oqkroiN[ji�UF�:p-q-_bsE_b[ji�UF�@\Vq�i�U�sEq-U-\^\V_�q�i�Z�P�U-\��oU�aVq#K

]`U-\�p-�4n:Z�adi�_bsE\ WoU�]`_badc#Zd �h+_bPNaVU�TdTkZJU�s�\V_bs4a�nos���_bsjU��oU�[j}oP�U(Z iNPL\���Z�t#iNa W�� p-�4n:Zd adi�_bsE\�WLi�"�p�TVU�s4adi�U�PNP�U-\ �oi�U�s�q-_b[j[gnosEU-\��7[ Z�i�\ U�PNP�U-\�TVp�t#i�\V\VU�s7a�P�� p��9_bPNnoadi�_bsYW�� U-\d}��-q-U-\�7iN�#Z�s4aVU-\�K���_b[j[IU�s4afWo�-\~P�_bTV\CsEU }:Zb\~\VU WoU�[ Z�sEWoU�Tf�7nEU�PNP�U R&noaCP�Z=Wop�[ Z�TVqVrEUjWoU-\\Vq�i�U�s4adi�):�7nEU-\ �7noi�p�akZ��oPNiNTVU�s7a^WoUCaVU�P�\ PNi�U�sE\ U�s4adTVU~P�U-\ [ Z�adrEp�[ Z�adi��7nEU-\^U�a P�Z TVp�Z�PNiNaVp}:Z�PN}:Z��oP�UfU�a��o}oPNnE\�}oTVp-q�i�\Vp�[IU�s7a��:P�� p-q-_bP�_bt#i�U��

]`_bTV\V�7nEUJ]`_badc#ZjU�a h+_bPNaVU�TdTkZ�adTkZ��bZ�iNPNP���TVU�s7a��@WoU~[ Z�soi���TVUgiNsEWop�}�U�sEWEZ�s7aVUF�:\dnoT�P�Z[I_7Wop�PNi�\kZ�adi�_bs WoU-\�TVU�P�Z�adi�_bsE\j}oTV_bi�U# �}oTVp-WEZ�aVU�noT��+iNP�\�Z�TdTdiN�9��TVU�s7a � WoU-\�q-_bsEq�PNnE\di�_bsE\�4noi`\VUCTVU-q-_bno}�U�s4aQ}:Z�Tdadi�U�PNP�U�[IU�s4a�K��C� U-\Va P�U~[���[IUf\dn��dU�a��4noi"Z�adadiNTkZ�P���Z�adaVU�s7adi�_bs WoU_ U�Z�TdP"\VnoT P�U~adTkZ-�#Z�iNP"WoUf]`_badc#Z��9U�TV\fM�X����LK

��U�adrE��[IU^p�akZ�s7a�P�U }�_biNs7a�WoU Wop�}:Z�Tda�WoU^q-U�adTkZ��bZ�iNP �4sE_bnE\�Z�PNP�_bsE\�sE_bnE\+}�U�sEqVrEU�T\dnoT P�U-\^Wop��9U�P�_b}o}�U�[IU�s4aV\J�7nEUgq-U-\^t#TkZ�sEWo\^sE_b[I\CWoUfP�Z \Vq�i�U�sEq-U Z�}o}�_bTdaV��TVU�s4a~WEZ�sE\P�UJq-_bs4aVU��LaVUgWoU~P�� U�s4adTVU WoU�nL� t#nEU�TdTVU-\%K

Page 10: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

M1� ��� ����� ���� ������

Page 11: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� �����������

� � �r� �� � � �� �������� 0 �� � � �

]+� U��L}oPNi�q�Z�adi�_bs WoU=P�ZY}oTVp-\kU�sEq-U WoU-\gR _bTd[IU-\ WoU=�7i�U P�U-\ }oPNnE\ q-_b[j}oP�U��oU-\ adi�U�s4aWEZ�sE\�WoU-\ P�_bi�\�Z�t#i�\V\ Z�s4a aV_bnoa�Z�noaV_bnoT�WoU^sE_bnE\��9}:Z�T U��oU�[j}oP�U P�ZJaVU�sEWEZ�sEq-UQt9p�sEp�TkZ�P�UWoU-\ U-\d}��-q-U-\ ��\VUJWop��9U�P�_b}o}�U�TQ\VU�P�_bsYnosEUJP�_bi�t9p-_b[Ip�adTdi��7nEUWZh0n<j�o kml�� h��_�v h�n<jpo U-\da P�Z�}�_b}onoP�Z�adi�_bs Z�n aVU�[j}E\ j � h�� P�Z�}�_b}onoP�Z�adi�_bs iNsoiNadi�Z�P�UeU�a l nosEUq-_bsE\dakZ�s4aVU~}�_#\diNadiN�9UC}oTV_b}oTVU[� P�� U-\d}��-q-U#K%O7noiN�#Z�s4a�nos�aVU�P�\VqVrEp�[ Z�LnosEUC}oP�Z�s7aVUJWo_bs4aqVr:Zb�7nEUft9p�sEp�TkZ�adi�_bs }�_bTdaVUgWoU�nL��t#TkZ�iNsEU-\^}oTV_�WLnoiNTkZ�iNaCU�s �7iNsot#a~Z�sE\ nos [jiNPNPNi�_bs WoU\VU-\"\VU�[!�oP�Z��oP�U-\���akZ�sEWLi�\��4n`� nos�q-_bno}oP�U�W�� p�P�p�}or:Z�s7aV\+Z�noTkZ�iNa�}oPNnE\8WoU��4noiNs ��U�[jiNPNPNi�_bsE\WoUJWoU-\Vq-U�sEWEZ�s4aV\~Z�}oTV�-\ q�iNsE��\di��-q�P�U-\�K

��U-\�p�adTkZ�sot9U-\�}oTVp-WLi�q�adi�_bsE\�TVU�[I_bs4aVU�s7a �f�JZ�T���iNsUH M�v��FX�@M�v#v�� J K�]"ZJTVp�Z�PNiNaVpCU-\dap��7i�WoU�[j[IU�s7a aV_bnoa�Z�noadTVU Z�\V_bs Zb\d}�U-q�a=sEU0\dnN�oiNa }:Zb\ WoU�t#TkZ�sEWo\�qVr:Z�sot9U�[IU�s7aV\W���Z�sosEp-UfU�s0Z�sosEp-U#KE]`Ufq-p�P��&�oTVUfs:Z�adnoTkZ�PNi�\daVU[�oTdiNakZ�sosoi��7nEUgZ sEp�Z�so[I_biNsE\r� not9pJnoadiNP�UWoUQq�Z�P�q�noP�U�T q-UQ}�_baVU�s7adi�U�P�}oTV_b}oTVUR�fP�Zgs:Z�adnoTVUfZd)Es WoUC[I_bs7adTVU�T �4nEUQP�U-\ U-\d}��-q-U-\ \VUPNiN�7TVU�s7aQU�s0TVp�Z�PNiNaVpI�jnosEUgPNnoadaVUf}�_bnoT^P�� U��Li�\daVU�sEq-U#K���U�adaVU WoU�Tdsoi���TVU q-_b[j}oTVU�sEW0sE_bs\VU�noP�U�[IU�s4a�P�Z q-_b[j}�p�adiNadi�_bs�U�s4adTVUJWoU�nL� U-\V}��-q-U-\��7[ Z�i�\�p�t4Z�P�U�[IU�s4a^P�Z Wop�}�U�sEWEZ�sEq-U�4n`� nosEU U-\d}��-q-UI}�U�noa Wop��9U�P�_b}o}�U�T�}:Z�T TkZ�}o}�_bTdaI��nosEU Z�noadTVU�_bn Z�nL��q-_bsEWLiNadi�_bsE\q�PNiN[ Z�adi��4nEU-\J}:Z�TfU��oU�[j}oP�U#K3� PNP�UITVU-q-_bno�7TVUjU�s R�Z�iNaJaV_bnoafq-Uj�4noi }�U�noaJ[IU�s:Zbq-U�TfP�Z\dnoTd�7i�UJW�� nos0iNsEWLiN�7i�WLn0_bn�WoUg\kZ�WoU-\Vq-U�sEWEZ�sEq-U#K

]"Z sEp-q-U-\V\diNaVpgWoU q-U�adaVUfPNnoadaVU }�_bnoTQP�� U��Li�\daVU�sEq-U p-�4noiNPNi��oTVU P�Zjs:Z�adnoTVU#K�� s U#"�U�a��aV_bnoaVUgnosEU \Vp�Tdi�U WoUK�:Z�TdTdi���TVU-\JU�[j}>�-qVrEU�s7aJP�U-\CU-\V}��-q-U-\CWoUgTVp�Z�PNi�\kU�TCP�U�noTC}�_baVU�s4adi�U�PWoU q�TV_bi�\V\kZ�sEq-U#K�]`U-\ TVU�P�Z�adi�_bsE\�U�s4adTVU q-U�PNP�U-\ 2q�i�\k_bs4a \VnoTd}oTVU�s:Z�s4aVU-\���}:Z�T U��oU�[j}oP�UF�P�U-\ qVr:Z�aV\��:U�s0q-_bs4adT�$bP�Z�s7aQP�UJsE_b[!�oTVUgWoUJ\V_bnoTdi�\��E�4noi`U�PNP�U-\Qq-_bs4adT�$bP�U�s7aQP�U-\^Z���U�iNPNP�U-\��U��LU�TVq-U�s7a�nosEUQiNs��EnEU�sEq-U~\dnoT P�U^sE_b[K�oTVU~WoU �:U�noTV\%K4]+� p-�7noiNPNi��oTVUCs:Z�adnoTVU�P�U-\Va�\dakZ��oP�UF�[ Z�i�\�P�UJ[I_biNsEWLTVUfqkr:Z�sot9U�[IU�s4aC}�U�noa P�U~}�U�TdadnoTp��U�TKS���L}`K`M1�dT2K

�C� U-\da"P�ZQ\VU�sE\di��oiNPNiNaVp�WoU+P�� p-�7noiNPNi��oTVU�s:Z�adnoTVU�PL�7noi4}�U�Td[IU�a�P�� p-q�P�_#\di�_bs WoU sE_bno�9U�PNP�U-\U-\d}��-q-U-\%K�e P�\dnPIa��7n`� nosEU�}�U�adiNaVU�}�U�TdadnoTp�:Z�adi�_bsI\VU }oTV_7WLnoi�\kU�}�_bnoT��4n`� nosEU�\diNadn:Z�adi�_bs

M#M

Page 12: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

M1� ��������� ��������� ��� � ��� ������� ��� � ��� � � ��� � ���

iNsEp-WLiNaVUCZ�}o}:Z�TkZ�i�\V\VU U�a��9\ViEq-U�adaVU \diNadn:Z�adi�_bs=\kU adTV_bno�9UL��adTVU R�Z-�9_bTkZ��oP�U �JP���Z�}o}:Z�TdiNadi�_bsW�� nosEU0sE_bno�9U�PNP�U�R _bTd[IU0WoUY��i�UF�+U�PNP�U0}�_bnoTdTkZ�\VU�Wop��9U�P�_b}o}�U�T�K � Z�i�\IP�Z s:Z�i�\k\kZ�sEq-UWoUYsE_bno�9U�PNP�U-\ U-\d}��-q-U-\I\���Zbq-q-_b[j}:Z�t#sEU iNsEp��7iNakZ��oP�U�[IU�s7a WoU�P�� U���adiNsEq�adi�_bs W���Z�noadTVU-\U-\d}��-q-U-\^U�aQP�� p-�4noiNPNi��oTVUgs:Z�adnoTVU�P�\kUJTVp�akZ��oPNiNaQWoUgPNnoi� �[���[IU#K@]"ZI}�_#\k\di��oiNPNiNaVpJ}�_bnoTQWoUsE_bno�9U�PNP�U-\fU-\V}��-q-U-\JW�� U��Li�\daVU�TJWop-q-_bnoP�UIWLn R Z�iNaJ�7nEU P�U sE_b[!�oTVUIW�� iNsEWLiN��i�WLnE\fW�� nosEUU-\d}��-q-U�s`� U-\Va }:Zb\ \VU�noP�U�[IU�s4a���_bTdsEp�}:Z�T P�� U�s7�7iNTV_bsosEU�[IU�s7a���[ Z�i�\�Z�nE\V\di^}:Z�T=P�U-\TVU�P�Z�adi�_bsE\ Z�nL� Z�noadTVU-\�U-\d}��-q-U-\�Z�PM�^_�vgP�Z PNnoadaVU }�_bnoT"P�� U��Li�\daVU�sEq-U�U-\da`P�Z^}oPNnE\"iNs7aVU�sE\VUF�P�U-\�qkr:Z�sEq-U-\ }�_bnoT WoU~sE_bno�9U�PNP�U-\^U-\d}��-q-U-\�WoU~�9_biNT�P�U��d_bnoT�\VU�TV_bs4a�P�U-\ }oPNnE\ t#TkZ�sEWoU-\�K�+PNP�U-\ \%��ZbWEZ�}oaVU�s4a�sE_bs \kU�noP�U�[IU�s4a9� P�U�noT�U�s4��iNTV_bsosEU�[IU�s4a�� [ Z�i�\�p�t4Z�P�U�[IU�s7a��P�U�noTV\ }oTV_bi�U-\��0��P�U�noTV\g}oTVp-WEZ�aVU�noTV\�� Z�nL��U-\d}��-q-U-\�Z-�9U-q=P�U-\V�4nEU�PNP�U-\ U�PNP�U-\�\V_bs4a�U�sq-_b[j}�p�adiNadi�_bsBU�a�� q-U�PNP�U-\I�4n`� U�PNP�U-\�sEU=R _bs4aI�7nEU�qz$baV_�9U�T�K e P�U-\Va�Wo_bsEq�sEp-q-U-\k\kZ�iNTVUWoUIq-_bsE\Vi�Wop�TVU�TfP�� U�sE\VU�[K�oP�U=WoU-\fTVU�P�Z�adi�_bsE\giN[j}oPNi��4n:Z�s7agnosEUIU-\d}��-q-U�}�_bnoTgU�s q-_b[I }oTVU�sEWLTVUgP�� p��9_bPNnoadi�_bs`K_ _bnoT�[ji�U�nL��q-_b[j}oTVU�sEWLTVU�q-U=q-_bsEq-U�}oa�WoUIPNnoadaVU=}�_bnoT P�� U��Li�\daVU�sEq-UF�"P�U=WEZ�T���i� soi�\daVU~Z�PNP�U�[ Z�sEW�� TdsE\da �CZbU-qkc9U�P H M�v�� { �@M�XLM�X J q�TVp�Z nosEU^sE_bno�9U�PNP�UC\Vq�i�U�sEq-UF��P�� p-q-_bP�_� t#i�UF�@Wo_bs4a^P�UgWo_b[ Z�iNsEUgW�� p�adnEWoUgWoU��#Z�iNa ��adTVUgP�U-\ TVU�P�Z�adi�_bsE\CWoU-\L��adTVU-\^��iN�#Z�s4aV\^U�s7adTVUU�nL�YU�a^Z-�9U-qJP�U�noT^[jiNPNi�U�n`KE�JZ�sE\�P�U-\ Z�sosEp-U-\gM�vFX���oWoU-\ ��_bakZ�soi�\daVU-\^Z�[Ip�Tdi�q�Z�iNsE\QWop# q�i�Wo��TVU�s7a^WoUJWop-\diNt#sEU�T�}:Z�T q-U~sE_b[ P�U-\�p�adnEWoU-\^Z�nL�o�4nEU�PNP�U-\ iNP�\�\VUCPNiN�7TkZ�i�U�s7a^q-_bsEq-U�T s:Z�s7a P�U-\j}oTV_�q-U-\V\dnE\jqkroiN[ji��4nEU-\ U�aj}orYL\di��7nEU-\IWoU-\I}oP�Z�s7aVU-\�K���U�P�Ze\�� iNsE\kq�TdiNa WEZ�sE\nosEU�aVU�sEWEZ�sEq-U^t9p�sEp�TkZ�P�UF� �CP�� p�}�_��4nEUF�9W���Z�}o}oPNi��7nEU�T P�U-\ [Ip�adrE_7WoU-\�W�� _��E\VU�Td�#Z�adi�_bs=U�aW�� U��L}�p�TdiN[IU�s7akZ�adi�_bs WoU~P�Zj}orYL\di��4nEU!� P�Z��oi�_bP�_bt#i�UWS���L}o}�M#MF�`M1�dT2K

OQ}oTV�-\jP�Z �CnEU�TdTVU0WoUBO�p-q-U-\V\di�_bs3��nosU�dU�nosEUYp�adnEWLi�Z�s4a U�s [Ip-WoU-q�iNsEU�WLn sE_b[WoU9O7aVU�}orEU�sBOQPSR&TVU-W `E_bTp��U-\BH M�v {#{ � M�X���� J \VU aV_bnoTds:Z �9U�TV\�P�� p�adnEWoUYWoU P�� roi�\daV_biNTVUs:Z�adnoTVU�PNP�U#K _ PNnE\�akZ�TVW:�9iNP@TVU�sEWLiNa }�_#\V\di��oP�U P�� _bno�9U�TdadnoTVUCW�� nosEUCp-q-_bP�U^W�� p�aVpCW�� roi�\daV_biNTVUs:Z�adnoTVU�PNP�UF�b}onoi�\�iNP7R _bsEWEZQP�U�]"Z���_bTkZ�aV_biNTVU�W����+akZ�a�W�� �Qi�\daV_biNTVU �~Z�adnoTVU�PNP�U�WoU�P���e PNPNiNsE_bi�\�Ke P�_�q-q�no}:Z WoU-\�}�_#\daVU-\�iN[j}�_bTdakZ�s4aV\��LiNP@R&noa�sE_bakZ�[j[IU�s4a(�fP�Zga���aVU~WLn �~p�}:Z�TdaVU�[IU�s4aWoU� �_�_bP�_bt#i�UfU�a W���� s7aV_b[I_bP�_bt#i�U�WoU~P�� |^soiN�9U�TV\diNaVpgWoUJP���e2PNPNiNsE_bi�\%Ke P�R noajP�� nosBWoU-\I}oTVU�[ji�U�TV\�� iN[j}�_#\VU�T \pL\daVp�[ Z�adi��7nEU�[IU�s4a nosEU+�:Zb\VU�\Vq�i�U�s7adi� ):�7nEU Z�nL��TVU-qkrEU�TVqVrEU-\~[IU�sEp-U-\~WEZ�sE\Q\k_bs Wo_b[ Z�iNsEU#K��CUg}oPNnE\��:\kU-\^adTkZ��bZ�nL�0\VU WLi�\ adiNsot#n:Z�i�U�s7agWoU-\JZ�noadTVU-\���q�Z�T~iNP"R�Z�i�\kZ�iNa]��U�Z�nEq-_bno}e}�_bnoT~p�akZ��oPNiNTfWoU-\QPNi�U�sE\fZ��9U-q�P�U-\WoU�Tdsoi���TVU-\JadrEp-_bTdi�U-\JU�s WEZ�aVU#KCO�U-\C}oTdiNsEq�iN}:Z�P�U-\f\V_bnoTVq-U-\~W�� iNsE\d}oiNTkZ�adi�_bs R noTVU�s4aJ�JZ�T ��iNs0U�a �QU�Tp��U�Tda(O7}�U�sEq-U�T�H M�v�����"M�X���� J �E}oroiNP�_#\V_b}orEU Z�sot#P�Z�i�\CWLn9']ea'~UJ\di��-q�P�U#KE�~U�JZ�T���iNs3� q#� U-\Va=p���i�WoU�[j[IU�s4a P�UY}oTdiNsEq�iN}�U WoUYP�� p��9_bPNnoadi�_bs TVp�t#i�U�}:Z�T P�Z \Vp�P�U-q�adi�_bss:Z�adnoTVU�PNP�U~�7noi@P�� iNs7aVp�TVU-\V\kZ��[ Z�i�\�iNP:�9_bnoP�Z�iNa�U�s Wop�aVU�Td[jiNsEU�T P�U-\ q�Z�nE\VU-\ U�a P�U-\ U#"�U�aV\�Ke P \VU�rEU�noTdakZ5� nos t#TkZ�sEWB}oTV_��oP���[IU5Z+\VU-\jp�adnEWoU-\=\VU�[K�oP�Z�i�U�s7a�Z�[IU�sEU�T }oPNnE\ WoUWLi�P q�noPNaVp-\C�4n`� U�PNP�U-\ s`� U�s�TVp-\V_bPN�#Z�i�U�s4a�K_ _bnoTLO7}�U�sEq-U�T���q-_b[j[IU^iNP@P�� p-q�TdiNa�WEZ�sE\�� .?2:,�G�� 2:.<;3=>.?8%476%,QU�s M�v#m���7iNP@U��Li�\daVUnosEUIR _bTVq-U Z��E\V_bPNnEU=�7nEUjsE_bnE\ sEUI}�_bno�9_bsE\ Z�}o}oTVp�rEU�sEWoU�T���[ Z�i�\ Wo_bs4a P�U-\fR _bTd[IU-\WoU R _bTVq-UQ�4nEU^sE_bnE\ q-_bsos:Z�i�\V\V_bsE\ Q P�ZJR _bTVq-U^�7iNakZ�P�UF�7\V_�q�i�Z�P�U^_bn=p-q-_bsE_b[ji��7nEULQ sEU\V_bs7a �7nEU�WoU-\ [ Z�soiSR U-\VakZ�adi�_bsE\�K7��U�adaVU R _bTVq-U Z��E\V_bPNnEU U-\da+q-_bsE\VakZ�s4aVU^WEZ�sE\+P�U aVU�[j}E\��

Page 13: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

M1�

P�U-\ Z�noadTVU-\�\dnN�oi�\V\kU�s4a�WoU-\gadTkZ�sE\ R _bTd[ Z�adi�_bsE\�K�]`U=}oTV_��oP���[IU�\VUITVp-\V_bnoa�Wo_bsEq��YP�U-\U���}oTdiN[IU�TI\V_bnE\ R _bTd[IUYW�� p-�7n:Z�adi�_bs3� P�U-\V�7nEU�PNP�U-\IWoU��7TkZ�i�U�s7aI[IU�adadTVUYU�s TVU�P�Z�adi�_bs P�Z[ Z�adi���TVUfU�a P�U~[I_bno�9U�[IU�s4a�K:OLZ�Wop#)EsoiNadi�_bs WoUfP�� p��9_bPNnoadi�_bs p�akZ�iNa P�Z�\dnoiN�#Z�s4aVUWZ

� �9_bPNnoadi�_bs�i�\8Z�s iNs4aVU�t#TkZ�adi�_bsj_�RE[ Z�adaVU�T Z�sEWgq-_bsEq-_b[jiNakZ�s4a WLi�\V\ViN}:Z�adi�_bs_�R�[I_badi�_bs��9WLnoTdiNsot���roi�qVr=adrEU [ Z�adaVU�T }:Zb\V\kU-\�R&TV_b[ Z�sIiNsEWoU#)EsoiNaVUF�4iNs q-_brEU�TVU�s4agrE_b[I_bt9U�sEU�iNaa aV_YZ=WoU#)EsoiNaVUF��q-_brEU�TVU�s7afrEU�aVU�TV_bt9U�sEU�iNaai��Z�sEWWLnoTdiNsot ��roi�qVr adrEUgTVU�akZ�iNsEU-W [I_badi�_bs nosEWoU�Tdt9_7U-\~Zj}:Z�TkZ�PNP�U�P�adTkZ�sE\ R _bT [ Z�adi�_bs�S��N�L}o} M { �"M � T2K

��U�adaVUjWop#)EsoiNadi�_bs�\VU �9_bnoP�Z�iNaf�#Z�P�Z��oP�UIZ�nE\V\di��oi�U�se}�_bnoT~P�� p��9_bPNnoadi�_bs s:Z�adnoTVU�PNP�U �4nEU}�_bnoT q-U�PNP�UfWLn \p�\VaV��[IUJ\V_bP�Z�iNTVUF�EWoU-\ q�iN��iNPNi�\kZ�adi�_bsE\^_bn�WoU-\Q\dadTdnEq�adnoTVU-\�t9p-_bP�_bt#i��7nEU-\S���L}o}�M1�d kM � ���~iNsot9\dP�Z�sEWYsE_bnE\^TVU�s4�9_bi�U Z�nL� � .<2:,�G � 2:.<;3=>.?8%476%,gWoU!O7}�U�sEq-U�T T2K

_ PNnE\�akZ�TVW:�uO7}�U�sEq-U�T�}onN�oPNi�Z�� �C6 � 2:.?;C=�.<8%476%,WE�� ].<E/47E��� U�s�M�v#m#m�oWEZ�sE\P�U-�4nEU�P�iNP�adTkZ�iNaVU sE_bakZ�[j[IU�s7agWoU-\C}oTV_��oP���[IU-\R�oi�_bP�_bt#i��7nEU-\�K>e2P�_��E\VU�Td�9U�WoU�nL� }oTdiNs q�iN}�U-\�K _ TVU�[ji���TVU�[IU�s4a��~}�_bnoT0P�ZB}oTVp-\VU�Td�#Z�adi�_bs W�� nosEU U-\d}��-q-UF�^iNP U-\da sEp-q-U-\V\kZ�iNTVU�4nEU P�U-\~R _bTVq-U-\~}oTVp-\VU�Td�#Z�adiN�9U-\f\V_bi�U�s4a U�s p-�4noiNPNi��oTVUIZ��9U-qjP�U-\~R _bTVq-U-\JWoU-\dadTdnEq�adTdi�q-U-\�K�CU�nL�Li���[IU�[IU�s4a���nosEUgR _bTdaVUg[I_bTdakZ�PNiNaVpj\���Zbq-q-_b[j}:Z�t#sEU�\V_bno�9U�s7aJW�� nosEUgR _bTdaVUgR U�Tdadi� PNiNaVpIaV_bnoa q-_b[j[IU=nosEUjR�Z�i��oP�U=[I_bTdakZ�PNiNaVp=\���Zbq-q-_b[j}:Z�t#sEU W�� nosEU R Z�i��oP�UIR U�TdadiNPNiNaVp#K0e2Pt9p�sEp�TkZ�PNi�\VU�TkZ q-U�adaVUQi�Wop-UCU�s nosYZ�s7akZ�t9_bsoi�\d[IUJU�s4adTVUC��i�U^iNsEWLiN��i�WLnEU�PNP�U~U�a�TVU�}oTV_�WLnEq# adi�_bs3�"Wo_bs4a iNP Wo_bsos:Z0nosEUIU���}oPNi�q�Z�adi�_bs Z�nosEU=U-\d}��-q-UF�"q-_b[j[IUIP�U-\ p�P�p�}or:Z�s4aV\��8�4noiWo_biN�9U�s4a�Wop�}�U�sE\VU�TK��U�Z�nEq-_bno} W�� p�sEU�Tdt#i�U }�_bnoT \VU�[ Z�iNs7aVU�soiNTjU�s �7i�UF�"sEUj}�U�no�9U�s4a�4n`� U�sfiNs7�9U-\dadiNT�}�U�n WEZ�sE\`P�U�noT"TVU�}oTV_�WLnEq�adi�_bs3����P�� iNs4�9U�TV\kU�W���Z�noadTVU-\�U-\d}��-q-U-\���q-_b[j[IUP�U-\CR _bnoTd[ji�\���Wo_bs7aJP�Z�\dnoTd�7i�U�q-_��oaVU�[I_biNsE\�K�]"Z�\kp�P�U-q�adi�_bses:Z�adnoTVU�PNP�UjsEU }�U�noa~}:Zb\��W�� U�PNP�U# �[���[IUF�"Z�not#[IU�s4aVU�TfP�Z R U�TdadiNPNiNaVpjU�afWLiN[jiNs4nEU�TfP�U-\]��U-\V_biNsE\C��iNakZ�nL� W�� nosEU�U-\ }��-q-U#K _ Z�T�q-_bs4adTVUF��P�� p��9_bPNnoadi�_bs }�U�noa8P�U R�Z�iNTVUF�%}:Z�T�U��LU�[j}oP�U�}:Z�T"P�Z^\d}�p-q�i�Z�PNi�\ Z�adi�_bs WoUq-U�TdakZ�iNsE\ iNsEWLiN��i�WLnE\��oq-_b[j[IUgqkrEU��fP�U-\ R _bnoTd[ji�\�KE|QsEUJU-\d}��-q-UJWoUg\dnoTd}oPNnE\ W�� p�sEU�Tdt#i�U� iNs4�9U-\dadiNT^U-\Va^Wo�-\�P�_bTV\ WLi�\d}�_bsoi��oP�U#K

�C� U-\da"}:Z�T�nos Z�Tdt#no[IU�s7a Wop-WLnEq�adiSR@�4nEU O7}�U�sEq-U�T Z�TdTdiN�#Z �QWoU+aVU�PNP�U-\�q-_bsEq�PNnE\di�_bsE\���4noi+[ Z�PNt#TVp P�U-\g�7nEU�P��4nEU-\ U��LU�[j}oP�U-\ �4noi�P�U-\gp�akZ�4Z�i�U�s4a sEUI}:Z�TkZ�i�\V\kZ�i�U�s7ag}:Zb\ p��7i� WoU�s4aVU-\R�I\VU-\Qq-_bs4aVU�[j}�_bTkZ�iNsE\�K3OLZIadrE�-\kUJR _bnoTdsoiNaQnosEU U���}oPNi�q�Z�adi�_bseWoUfP�Z \Vp�P�U-q�adi�_bss:Z�adnoTVU�PNP�U U�s aVU�Td[IU-\~WoUgP�_bi�\~\diN[j}oP�U-\CU�aCq-UgR noa^PM�=\V_bs0}oTdiNsEq�iN}:Z�P�iNs4aVp�T���a�K>`E_bTp��U-\ZbWLrEp�TkZ��jq-U�adaVUfadrEp-_bTdi�U \VU�P�_bs�P�Zb�7nEU�PNP�UJP�� U-\d}��-q-Uf\dnoTd��iNaQZ�n0Wop�}�U�sE\^WoUJP�� iNsEWLiN��i�WLn`K_ _bnoT�PNnoi �9nosEUQU-\d}��-q-U^U�a�\VU-\�U�sosEU�[ji�\�_bs7a�WoU-\�iNs4aVp�T���aV\�q-_b[j[gnosE\�K� s U#"�U�a��7\diEnos}oTVp-WEZ�aVU�noTf�9_biNaf\VU-\~}oTV_bi�U-\f\VU�TkZ�TVp#):U�T Z�n 2WoU�PM��W�� nosEUIq-U�TdakZ�iNsEUjPNiN[jiNaVUF��iNP \VU�[IU�aU�s�WEZ�sot9U�T�K

��i�U�s��4nEU~P���Zb\d}�U-q�a WoU~P�Z s:Z�adnoTVUJsEUJ\dnN�oi�\k\VUC}:Zb\ WoUJqVr:Z�sot9U�[IU�s7aV\QWLTkZb\dadi��7nEU-\W�� nosEU Z�sosEp-UI�jP���Z�noadTVUF�@P�UfsE_b[K�oTVU W�� iNsEWLiN��i�WLnE\CW�� nosEU U-\d}��-q-Ufs`� U-\daC�4nEUgadTV�-\^TkZd TVU�[IU�s4a q-_bsE\dakZ�s7a�� iNP _#\Vq�iNPNP�UYWLn R�Z�iNaIWoU-\�iNs4aVU�TkZbq�adi�_bsE\ �4n`� U�PNP�UY\VnN�oiNa�Kr� Z�PNt#TVp�P�Uqz$baVp iN[j}oP�Zbq�Z��oP�U��7nEUITVU��i��a P�Z�PNnoadaVU=}�_bnoT P�� U���i�\VaVU�sEq-UF�"P�U=}oTdiNsEq�iN}�U WoUIP�Z \Vp�P�U-q# adi�_bs s:Z�adnoTVU�PNP�U _�U�no��TVUY}�_bnoT P�U9�oi�U�sBq-_b[j[ nos3��q�Z�T�Z�iNsE\di P�Z s:Z�adnoTVU0aVU�sEW �9U�TV\nos�p-�4noiNPNi��oTVU#KN`E_bTp��U-\�TVU�[ Z�TV�7n:Zj�7nEUJ\di�P�Z s:Z�adnoTVU~aVU�sEWY�9U�TV\ nos�_bTVWLTVUR��p�sEp#):�4nEU

Page 14: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

M { ��������� ��������� ��� � ��� ������� ��� � ��� � � ��� � ���

}�_bnoT�aV_bnE\��LsE_bnE\�}�_bnoTdTV_bsE\^q-U�TdakZ�iNsEU�[IU�s7aCZ�adaVU�sEWLTVUR�oi�U�s0W���Z��bZ�s7akZ�t9UgWoU~P�� r7no[ Zd soiNaVpF�Lq�Z�T �fP�ZgaVU�sEWEZ�sEq-UCs:Z�adnoTVU�PNP�UJ\���Zd� _bnoaVU�s7a�P�U-\ U#"�_bTdaV\ WoU^P�����adTVUCr4no[ Z�iNsUS��N�7}o}M � kM%� � �~iNsot9\dP�Z�sEW q�iNaVU�p�t4Z�P�U�[IU�s7a � �C6�� 2:.?;C=�.<8%476%,RE�� ].?E04YE��� WoU O7}�U�sEq-U�T T2K

_ _bnoT~P���Z�[Ip�Tdi�q�Z�iNs3��P�ZIa � qkrEU WLn \Vq�i�U�s7adi�):�4nEU�p�akZ�iNaJWoUgq-_b[j}oTVU�sEWLTVUjq-_b[j[IU�s7aP�U-\�TVU�P�Z�adi�_bsE\QWoU~}oTVp-WEZ�adi�_bs0U�a^WoUJq-_b[j}�p�adiNadi�_bs�}�_bno�bZ�i�U�s7a [ Z�iNs4aVU�soiNTQP�� r:Z�Td[I_bsoi�UWEZ�sE\ P�Z s:Z�adnoTVU0U�a=Z��9_biNT \dnoT U�PNP�UYnosEU�iNs��EnEU�sEq-U9��p�sEp#):�4nEU#K+O q-UY[I_b[IU�s7a �PM��U�s M�vFX { � `E_bTp��U-\�R noaYP�� nos WoU-\�}oTVU�[ji�U�TV\D� \VUe�7n:Z�PNi�):U�T�W�� p-q-_bP�_bt#i�\daVU#KQ��U�nL� 2q�iiNsosE_%�9��TVU�s4aJU�s�iNsE\di�\dakZ�s7a^\dnoT P�� p�adnEWoU WoU-\ q-_b[j[gnos:Z�noaVp-\C}oPNnoa�$baC�4nEUf\VnoT q-U�PNP�UgWoUP�� iNsEWLiN��i�WLn3�4U-\V\ Zz4Z�s4a Z�iNsE\di:WoU Wop-q-_bno��TdiNT�q-U^�4noiEWo_bsos:Z�iNa �~P�Z~s:Z�adnoTVU \kZf\dadTdnEq�adnoTVU\di`q-_brEp�TVU�s7aVU#K

] � p�adnEWoUfWoU-\�_bTdt4Z�soi�\d[IU-\�P�U�noT R noa�sEp�Z�so[I_biNsE\�noadiNP�U#KNe P�\ IR�Z�i�\kZ�i�U�s4a�TVp�R p�TVU�sEq-UP�_bTV\V�7n`� iNP�\�_��E\VU�Td�bZ�i�U�s7a�WoU-\�Z�s:Z�P�_bt#i�U-\�K _ Z�T�U��oU�[j}oP�UF��`oTVU-WoU�Tdi�q�� K#��P�U�[IU�s4aV\ H M�v9� { �M�X { � J �"P�UjP�U�ZbWoU�T WoU-\��+akZ�aV\ 2|^soi�\ U�s [ Z�adi���TVU W�� p-q-_bP�_bt#i�U=�9p�t9p�akZ�P�U=Z�n�Wop&�onoa WLn\di��-q�P�UF�8iNs��EnEU�sEq-p PNnoi�Z�nE\V\Vi }:Z�T�O7}�U�sEq-U�T��8�9_�4Z�iNa�P�U-\ q-_b[j[gnos:Z�noaVp-\Iq-_b[j[IU WoU-\_bTdt4Z�soi�\d[IU-\ q-_b[j}oP�U��oU-\g�7noi+Wo_biN�9U�s4a \kUIWop��9U�P�_b}o}�U�T���Z�TdTdiN�9U�T!��[ Z�adnoTdiNaVp=U�agWLi�\ }:Z�TkZ � adTVU#K�]`U-\~}�_b}onoP�Z�adi�_bsE\JWo_biN�9U�s4a Z�iNsE\di�Z�adaVU�iNsEWLTVU�nosep�akZ�aJW�� p-�7noiNPNi��oTVUjq-_bTdTVU-\ }�_bsEWEZ�s4a]� P�U�noTQZ�}�_bt9p-U#K

]8ZY\dnEq-q-U-\V\di�_bs�WoU-\fU-\d}��-q-U-\fU-\da Z�P�_bTV\JTVp�t#i�UI}:Z�TfnosEUIq-_b[j}�p�adiNadi�_bs�U�s4adTVU=WoU-\}oP�Z�s7aVU-\ \diN[jiNP�Z�iNTVU-\ Zz4Z�s7a�P�U-\�[���[IU-\ ��U-\V_biNsE\%K7]`U~Wop��9U�P�_b}o}�U�[IU�s4a^\%��Zbq-q-_b[j}:Z�t#sEUW�� nosEUjWLiN[jiNs7noadi�_bs�WoU P�Z=q-_b[j}�p�adiNadi�_bs��7noi�q-�-WoUj\ ZI}oP�Zbq-U��=nosEU roi�p�TkZ�TVqVroi�\kZ�adi�_bsWoU-\�U-\d}��-q-U-\�K�|^sEU�}oPNnE\�t#TkZ�sEWoU WLiN�9U�TV\diNaVp U-\da�Wo_bsEq \p�sE_bsY�[IU�WoU�[I_biNsEWLTVU q-_b[j}�p# adiNadi�_bs U�a W�� nos Wop��9U�P�_b}o}�U�[IU�s4a�}oPNnE\+Z-�#Z�sEq-p#K7��U�adaVU���i�\di�_bsjWoU-\+qVrE_#\kU-\+U�noa+q-_b[j[IUZ��bZ�s7akZ�t9U W���Z�adadiNTVU�T+P���Z�adaVU�s4adi�_bs WoU-\��oi�_bP�_bt#i�\daVU-\+\dnoT�P�U-\�TVU�P�Z�adi�_bsE\�q-_b[j}oP�U��LU-\+U�s7adTVUnos�_bTdt4Z�soi�\d[IUJU�a \k_bs�U�s7�7iNTV_bsosEU�[IU�s7a�K:]"Z��oi�_bP�_bt#i�UF�oq-_b[j[IUfP�� p-q-_bP�_bt#i�UF�oWoU��9U�s:Z�iNaW�s:Z�[ji��4nEUgU�a^U���}�p�TdiN[IU�s4akZ�P�U S���L}o}�M%� �`M�vdT2K

��r:Z�TdP�U-\ ��roTdi�\VaV_b}orEU�TIO�WEZ�[I\9H M�v9�G���M�X �F� J adTkZ��bZ�iNPNP�Z Z�nL� qz$baVp-\�WoU�`E_bTp��U-\}�U�sEWEZ�s4a WoU�nL� Z�sE\�K�e P��9_bnoP�Z�iNa=�4nEUYq-U�adaVU�sE_bno�9U�PNP�U�\kq�i�U�sEq-UF��P�� p-q-_bP�_bt#i�UF� Z�iNaIWoU-\�:Zb\VU-\ \kZ�iNsEU-\�KFe PLp�akZ�iNa+\V}�p-q�i�Z�P�U�[IU�s7a�q-_bsEq-U�TdsEp�}:Z�T P�U�Wop-q�Z�P�Z�t9U U�s7adTVU�P���Z���_bsEWEZ�sEq-UWoU�R�Z�iNaV\`U�a`P�Z�TkZ�TVU�aVp�WoU-\�adrEp-_bTdi�U-\"WEZ�sE\"P�U+Wo_b[ Z�iNsEU#K%�JZ�sE\�P�U �onoa"WoU�}�U�Td[IU�adadTVU�\V_bs��_bs Wop��9U�P�_b}o}�U�[IU�s7a��`iNP�adTkZ-�#Z�iNPNP�Z9� nosEUIZ�}o}oTV_�qVrEUIWoUjq-U�adaVU�sE_bno�9U�PNP�U WLi�\Vq�iN}oPNiNsEUU�aJ}onN�oPNi�Z�\VU-\~i�Wop-U-\JWEZ�sE\��I13.<AC6^G:E G �36��uG:13A E���I;3. � -04U* =>E/47E��� U�sM�XLM1�LKO�_bs }�_biNs4a�WoU���nEUF�+P�Z�2[Ip�adrE_�WoU W7s:Z�[ji��7nEU �0aVU�PNP�U �4n`� iNPQP���Z�}o}�U�P�Z�iNa���p�akZ�iNaWEZ�sE\ P���Z�iNTjWLn aVU�[j}E\�K 8_bnoac U-\da roi�p�TkZ�TVqVroi�\VpYU�s \pL\daV��[IU-\��8WoU=P�� p�P�U-q�adTV_bs � P�Zq-_b[j[ nos:Z�noaVp U�s }:Zb\k\kZ�s4a�}:Z�T�P�U-\ iNsEWLiN�7i�WLnE\�K |Qs \��\daV��[IU \VU q�Z�TkZbq�aVp�Tdi�\VU }:Z�T\kZ�q�Z�}:Zbq�iNaVp ��TVp�Z�t#iNTI�YnosEU }�U�TdadnoTp�:Z�adi�_bs U�a!�Y[ Z�iNs7aVU�soiNT nos p�akZ�agW�� p-�7noiNPNi��oTVU#K��_b[j[IU9`:_bTp��U-\jU�a=��P�U�[IU�s7aV\��+O�WEZ�[I\ Z \dnN�oi P�� iNs��EnEU�sEq-U�WoUBO7}�U�sEq-U�T���[ Z�i�\jiNP

Page 15: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

M �\�� p�akZ�iNa"p�t4Z�P�U�[IU�s4a8iNs4aVp�TVU-\V\kp ��P�Z�qkroiN[ji�U+}orYL\di��7nEU#K�e P#�9_bnoP�Z�iNa8q-_b[K�oiNsEU�T8P�U+q-_bsEq-U�}oaW�� _bTdt4Z�soi�\d[IU q-_b[j}oP�U��oU U�a WoU-\g_bnoadiNP�\ q-_b[j[IUIP�U-\g\��\daV��[IU-\���P�U-\ �EnL� W�� p�sEU�Tdt#i�UF�P���Zbq�adi�_bs U�a�P�Z~TVp�Zbq�adi�_bs3��}�_bnoT\)Es:Z�P�U�[IU�s4a�Z�}o}oPNi��4nEU�T �JP�� p-q-_bP�_bt#i�UCWoU-\ TVp-\VnoPNakZ�aV\�WoUP�ZgqkroiN[ji�U~}orYL\di��4nEUCU�a�q-_bsE\dadTdnoiNTVUCnosEUCadrEp-_bTdi�UQ}�_bnoT U�s)Es�_bTVWo_bsosEU�T q-UQsE_bno�9U�Z�nWo_b[ Z�iNsEUf�4noi`U�s Z-�#Z�iNa akZ�s4aL��U-\V_biNs`K

X#nE\V�7nEU# �PM��EP�U-\C\dn��dU�aV\^W�� p�adnEWoU p�akZ�i�U�s4aCP�U-\QU-\d}��-q-U-\��Eq#� U-\Va �d 2WLiNTVUgP�U-\QTVp-\dnoPNakZ�aV\WoU P�� p��9_bPNnoadi�_bs`KL�~p-\V_bTd[ Z�i�\ q-UQWoU��#Z�iNa���adTVUQP�U-\�}oTV_7q-U-\V\VnE\�WoU^P�� p��9_bPNnoadi�_bs`K��Qs�\�� iNs aVp�TVU-\V\kZ�iNa^Z�nL��R _bsEq�adi�_bsE\ }oPNnoa�$baQ�7n`��Z�n�R _bTd[IU-\Q�4n`� U�PNP�U~TVU��i��a�K@|^sEUCR _bi�\ U�sEq-_bTVUF�oP�ZWop�[ Z�TVqVrEU�W�� O�WEZ�[I\�WoUjR _7q�Z�PNi�\kZ�adi�_bs \dnoT P�U-\g}oTV_�q-U-\V\dnE\�Z�n�Wop�adTdiN[IU�s7aIWLn�TVU-\daVUs`� U-\da+}:Zb\�\diNsot#noPNi���TVU �~P�� p�}�_��4nEU#K _ U�noa a��adTVU~Z�PNP�Z�iNa 2U�PNP�U^adTV_b}=P�_biNs3�9[ Z�i�\�q-U�adaVU sE_bn �9U�PNP�UjR�Z�b-_bs�WoUI}�U�sE\VU�T���Z�}oTV�-\ Z��9_biNT }:Zb\V\ Z��oP�U�[IU�s4a iNs��EnEU�sEq-p P�Z�}orYL\di��4nEU=U�agP�ZqVroiN[ji�Ufq-_b[j[IU�s%b�Z�iNaR��\�� p�akZ��oPNiNT^U�s9�oi�_bP�_bt#i�U#K

��_b[j[IUJW���Z�noadTVU-\ �oi�_bP�_bt#i�\daVU-\^Z��bZ�s7a�PNnoi �oO�WEZ�[I\�\�� iNsE\d}oiNTkZ�WoUCP�Z t9p-_bP�_bt#i�U#KE]`UT�$bP�U0iN[j}�_bTdakZ�s4a�WoU-\=}oTV_7q-U-\k\dnE\�U�s p-q-_bP�_bt#i�U p�akZ�iNa�aV_bnoa�� R�Z�iNa U�s Zbq-q-_bTVW Z-�9U-qP�Zj��i�\di�_bs0t9p-_bP�_bt#i��7nEU#K�]+� O^[Ip�Tdi�q�Z�iNseWop�}:Zb\V\ ZI\VU-\^}oTVp-Wop-q-U-\V\VU�noTV\~U�s TkZ�}o}oTV_�qVr:Z�s7aP�� p��9_bPNnoadi�_bs WoU P�� p�TV_#\di�_bs3� q-_b[j}:Z�TkZ�s4a�P�U }oTVp-WEZ�aVU�noT��4noiCWop��9_bTVU U�a�Zb\V\ViN[jiNP�U \kZ}oTV_bi�UK� P�� U�Z�n0�7noi`p�TV_7WoUfU�a U�sot#P�_bnoadiNaQP�U~TVU�PNi�U�RLS���L}o}�M�Xd ��LM#T2K

O�P�� iNsE\dakZ�T�WoU=q-U�nL���7noi �8q-_b[j[IUY��P�U�[IU�s4aV\���q-_bsE\Vi�Wo��TVU�s4a�P�Z�s:Z�adnoTVU q-_b[j[IUnos=_bTdt4Z�soi�\d[IUQq-_b[j}oP�U��LUC_bn=WoU q-U�nL� �7noi �4q-_b[j[IU `E_bTp��U-\��9P�Zf�9_bi�U�s4a Wo_baVp-U^W�� nosEUt#TkZ�sEWoU�\VU�sE\Vi��oiNPNiNaVp Z�nL� qVr:Z�sot9U�[IU�s4aV\��bP�Z [Ip�adrE_�WoU W7s:Z�[ji��7nEU�W�� O�WEZ�[I\ U-\Va"}oPNnE\�4n`� nosEU�\diN[j}oP�U R�Z�b-_bseW�� p�adnEWLi�U�TJP�Z �oi�_bP�_bt#i�U#K"��U-\~adTV_bi�\Jq-_bsEq-U�}oadi�_bsE\f}�U�Td[IU�adaVU�s4aW�� U�s4��i�\kZ�t9U�T nosEUfq-_b[j[ nos:Z�noaVp _bTdt4Z�soi�\Vp-UfTVp�Z�t#i�\V\kZ�s4aQq-_b[j[IUfnosYaV_bnoa�K

��U �7noi�Z adTkZ�iNaR�IP�ZIPNnoadaVU }�_bnoT^P�� U��Li�\daVU�sEq-UgR Z�iNaC}:Z�Tdadi�U WLn Wo_b[ Z�iNsEU�W�� p�adnEWoUWoU-\ p-q-_bP�_bt#i�\VaVU-\�� �4noi�U-\dajWo_bsEq adTV�-\��#Zb\daVU#K��CU PM� Wop-q-_bnoP�U�P�Z WLi�P q�noPNaVpYW�� Z�} }oPNi��4nEU�TInosEU�adrEp-_bTdi�U0\dadTdnEq�adnoTVp-U#K e2PL Z�+W�� nosEU0}:Z�Tda�� P�U-\ \dadTdnEq�adnoTVU-\=U�s }oP�Zbq-UF�W���Z�noadTVU=}:Z�Tda���P�U-\ }oTV_�q-U-\V\dnE\��4nEU \dnN�oi�\V\kU�s4a�P�U-\jZbq�aVU�noTV\j�4noi p��9_bPNnEU�s4aIWEZ�sE\�q-U-\\dadTdnEq�adnoTVU-\DZ P�U-\ U-\d}��-q-U-\�K�]+� p�adnEWoU WoU-\ t#TV_bno}�U-\ W�� _bTdt4Z�soi�\d[IU-\WQ q-_b[j[ nos:Z�n aVp-\��@\k_7q�i�p�aVp-\���Zb\V\k_7q�i�Z�adi�_bsE\~U�a]�oi�_7q-p�sE_#\kU-\ Q U-\VaCp�t4Z�P�U�[IU�s7afiN[j}�_bTdakZ�s7aVUjU�aCR Z�iNa}:Z�Tdadi�U WoUIP�� roi�\VaV_biNTVUIs:Z�adnoTVU�PNP�U#K0� s)Es3�"P���Z�s:Z�P�L\VU=WoUIP�� p��9_bPNnoadi�_bs WoU q-U-\ft#TV_bno}�U-\WEZ�sE\fP�UIaVU�[j}E\ U�agP�� U-\V}:Zbq-U q-_bsEq-U�TdsEU=P�ZYt9p-_bt#TkZ�}oroi�U U�agP�Z�t9p-_bP�_bt#i�U#K ��U \V_bs7agPM��4nEU�P��7nEU-\ �nosEU-\ WoU-\ WLiN�9U�TV\kU-\�R�Zbq-U�adaVU-\ WoUJP�� p-q-_bP�_bt#i�U#KO7i P�U-\ p-q-_bP�_bt#i�\daVU-\ }�_bno�#Z�i�U�s4a�q-_b[j}oTVU�sEWLTVU P�ZYs:Z�adnoTVUF�8iNP�\gWoU���TkZ�i�U�s4ac��adTVU U�s[IU-\dnoTVUF�`aV_bnoaVU }oTV_b}�_bTdadi�_bs�t4Z�TVWop-UF�"WoUIP�ZYt9p�TVU�Tg}�_bnoTgP�Uc�oi�U�s WoUjP�Z�\V_�q�i�p�aVpF�`U�s}:Z�Tdadi�q�noPNi�U�TIq-_bsEq-U�Tds:Z�s4ajP���Z�t#Tdi�q�noPNadnoTVU�U�a P�Z }��-qkrEU#K ��U�\V_bs7ajq-U-\ }oTV_b[IU-\k\VU-\��4noi� nE\dadi�):��TVU�s7a�P�U-\ [I_�9U�sE\�[ji�\�U�s _�U�no��TVU }�_bnoT P�U�Wop��9U�P�_b}o}�U�[IU�s4aIWoU P�� p-q-_bP�_bt#i�U#KO�WEZ�[I\fZ�PNP�Z }oPNnE\~P�_biNs Z�iNP�P�Z=}oTVp-\VU�s4akZ�q-_b[j[IU�P�U�qz$baVpjZ�}o}oPNi��4nEpjWoU P�Z �oi�_bP�_bt#i�UF�Zd� _bnoakZ�s7a��7n`� U�PNP�UQ\kU�TVp�R p�TVU~Z�nL� \Vq�i�U�sEq-U-\�s:Z�adnoTVU�PNP�U-\ U�a�\V_�q�i�Z�P�U-\+P�_bTV\k�4nEUQiNP:P�U R Z�noa�K

��U�adaVUCiNsE\di�\dakZ�sEq-U~\dnoT�aV_bnoa q-UC�7nEUQP�� p-q-_bP�_bt#i�UC}�U�noa Z-�9_biNT�W���Z�}o}oPNi��4nEpfZ q-U�TdakZ�i� sEU�[IU�s4a�R Z��9_bTdi�\Vp~\V_bs�U-\V\V_bT��7[ Z�i�\�q-U�P�Zg\�� U-\Va�}oTV_��:Z��oP�U�[IU�s7a�R Z�iNa�Z�nYWop�adTdiN[IU�s4a WoUq-U�TdakZ�iNsE\fp�P�Z�sE\~}oPNnE\~adrEp-_bTdi��4nEU-\%K`]+� U�s4aV_b[I_bP�_bt#i�UF�"}:Z�TfU��LU�[j}oP�UF��R _�q�Z�PNi�\kZ�iNa]��U�Z�n

Page 16: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

M�m ��������� ��������� ��� � ��� ������� ��� � ��� � � ��� � ���

q-_bno}0W���Z�adaVU�s7adi�_bs3�@Z�P�_bTV\ �7nEU~P�U-\�}oTV_��oP���[IU-\ ��_7_bP�_bt#i��7nEU-\ }oPNnE\�t9p�sEp�TkZ�nL�YiNs7aVp�TVU-\ \kZ�i�U�s7a WoU [I_biNsE\gU�s�[I_biNsE\%K8��U�P�Z�R�Z�iNPNPNiNagr77}�_badrEp-�4nEU�T P�UIWop��9U�P�_b}o}�U�[IU�s4a�adrEp-_� Tdi��7nEUfWoUJq-U�adaVUJsE_bno�9U�PNP�U \Vq�i�U�sEq-UF�EU�a�}:Z�T PM�d �[���[IU \V_bs0Z-�9U�soiNT�K

]"U-\ �oi�_bP�_bt#i�\daVU-\~WoUgaVU�TdTkZ�iNs �9_bnoP�Z�i�U�s7a���WoUgP�U�noTQqz$baVpF�@[ Z � adTdi�\VU�TC}:Z�T R Z�iNaVU�[IU�s7anos }�U�adiNa^\kU�t#[IU�s4a~WoUJs:Z�adnoTVU#K@|QsEUJadrEp-_bTdi�U t9p�sEp�TkZ�P�UfadTV_b} p�P�_biNt#sEp-U WoUJP�ZITVp�Z�PNiNaVpsEUCP�U-\ iNs4aVp�TVU-\k\kZ�iNa�t#nE��TVU#K:��U�}�U�sEWEZ�s7a��EZ�nE\V\di�}�U�adiNa \k_biNa �iNP �7nos�\kU�t#[IU�s4a WoUCs:Z�adnoTVUTVU-\daVUgU���adT���[IU�[IU�s7aCq-_b[j}oP�U��oUgU�a P�Ug\VU�noP`[I_�9U�s W���Z-�#Z�sEq-U�TCU-\Va^WoUJ}oTV_�q-p-WoU�T }:Z�T\diN[j}oPNi�):q�Z�adi�_bsE\��:q-UJ�7noi"\VU~TkZ�}o}oTV_�qVrEUJt#TkZ�sEWoU�[IU�s7aCW�� nos Z�}o}oTV_�qVrEUgadrEp-_bTdi��4nEU#K

]8Z sEp-q-U-\V\diNaVp~W�� nosEUCadrEp-_bTdi�UJ\dadTdnEq�adnoTVp-UJp�akZ�iNa�q-U�TdakZ�iNsEU#Ko]`U~}oTV_��oP���[IUJWoU�[IU�n TkZ�iNa q-_b[j[IU�s4a P���Z�}o}oPNi��7nEU�T�Kb]+� p-q-_bP�_bt#i�U Z�PNP�Z�iNa \dnN�oiNT�nos�Wop��9U�P�_b}o}�U�[IU�s4a+WoU�}oPNnE\ U�s}oPNnE\g[ Z�adrEp�[ Z�adi��4nEU#K8O P�� p�}�_7�7nEUF�`_bs�\kU�s4akZ�iNagWop ���Y�7n`��Z�nEq�nosEUjadrEp-_bTdi�UI}:Z�T R�Z�iNaVUsEUJ}�_bnoTdTkZ�iNa ��adTVU p�P�Z���_bTVp-UF�@�4n`� iNP`R Z�nEWLTkZ�iNaC\VUfq-_bs4aVU�s7aVU�TCW���Z�}o}oTV_���iN[ Z�adi�_bsE\~}�_bnoTU��L}oPNi��4nEU�T^nosY[I_bsEWoUgWoU~}oPNnE\^U�sY}oPNnE\Qq-_b[j}oP�U��LUWS���L}o}O����[���dT2K

Page 17: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� ����������� �

� ���u� � � � � ��� � �

O^n Wop&�onoa WLn \di��-q�P�UF�8P�� p-q-_bP�_bt#i�U=s:Z��7iNt#n:Z�iNa�U�s4adTVU P�� p�adnEWoU WoU P�� _bTdt4Z�soi�\V[IU=U�aq-U�PNP�UfWoU-\^q-_b[j[ nos:Z�noaVp-\CW�� U-\V}��-q-U-\��:\kZ�sE\^\kZ��9_biNTQ_�v0\kUf\diNadnEU�T^U�s7adTVUgq-U-\^WoU�nL��WLi� TVU-q�adi�_bsE\=q-_bs7adTkZ�iNTVU-\�K ]`U9��U-\V_biNs W�� _bTVWLTVU \VU�R Z�i�\kZ�iNa \VU�s4adiNT WEZ�sE\=q-U�adaVU sE_bno�9U�PNP�UWLi�\Vq�iN}oPNiNsEU#Ko]"Z ��i�\di�_bs�WoU-\�p-q-_bP�_bt#i�\daVU-\^p�akZ�iNa U-\k\VU�s4adi�U�PNP�U�[IU�s7aQ�7n:Z�s4adiNakZ�adiN�9U#K _ _bnoT �4nE_bi�U�a�q-_b[j[IU�s7a�nosEU�}�_b}onoP�Z�adi�_bs �#Z�Tdi�U# �a 2U�PNP�U U�s sE_b[!�oTVU�� �C� U-\dar� P�� iN[j}�_bTdakZ�sEq-UWoUJq-U-\ �4nEU-\Vadi�_bsE\^�7nEU~P�� p-q-_bP�_bt#i�UgWo_biNa \V_bs0Wop��9U�P�_b}o}�U�[IU�s4a�K

]+� p��9_bPNnoadi�_bs WoU-\ aVU-qVrosoi��7nEU-\jWoU \daV_�qVc#Z�t9U=U�a�WoU=TVp�R&TdiNt9p�TkZ�adi�_bs Z�n Wop&�onoa WLn\di��-q�P�U WoU�Tdsoi�U�TfU�noaC}�_bnoT~q-_bsE\kp-�4nEU�sEq-U P�U��oTkZb\V\kZ�t9U WoU-\~iNsE\VU-q�aVU-\Cs4noi�\di��oP�U-\[�=adTkZd �9U�TV\~P�U-\Jq-_bs4adiNsEU�s4aV\%K _ _bnoTfq-U�adaVU�TkZ�i�\k_bs3��P�� U�s4aV_b[I_bP�_bt#i�UIU�a~P�� p-q-_bP�_bt#i�UIU�s t9p�sEp�TkZ�P��p�sEp#):q�i���TVU�s7a�W�� nosEU�Z�adaVU�s7adi�_bsjZbq-q�TdnEU#Kb]"Z^}oTVU�[ji���TVU�t#nEU�TdTVU�[I_bsEWLi�Z�P�U�TVU�sEWLiNa��bU�PNP�UZ�nE\V\di �@P�U-\fZ-�#Z�sEq-p-U-\JU�s p-q-_bP�_bt#i�U�adTV�-\CiN[j}�_bTdakZ�s4aVU-\��`P���Z�t#Tdi�q�noPNadnoTVUjp�akZ�s4aJ}:Z�Tdadi�q�n PNi���TVU�[IU�s4aQ\k_bPNPNi�q�iNaVp-U#K� Z�PNrEU�noTVU�nE\kU�[IU�s4a���P�U adTkZ��bZ�iNP�WoU-\�p-q-_bP�_bt#i�\VaVU-\�WoUQP�� p�}�_��7nEU�P�U�noT�TVp��9p�P�ZgU-\V\VU�s adi�U�PNP�U�[IU�s4aJP�� p�aVU�sEWLnEUIWoU P�U�noT~iNt#sE_bTkZ�sEq-U R Zbq-UjZ�nL�0P�_bi�\~s:Z�adnoTVU�PNP�U-\%K�]+��Z�}o}:Z�TdiNadi�_bsW�� iNs4�#Zb\di�_bsE\JW�� iNsE\VU-q�aVU-\JsEU!)EaJ�4n`��Zd�d_bnoaVU�TK�=P�U�noTJWop-\kZ�TdTV_bi�K�O�P�ZW)EseWLnU'(e 'C��[IU\di��-q�P�UF�bP�� iNs4adTV_�WLnEq�adi�_bs WoU P�Zfq-_7q-q�iNsEU�PNP�U U�s�� Z�PNiSR _bTdsoi�U U�noa�WoU-\ TVp-\dnoPNakZ�aV\ \V}�U-q�akZbq�n P�Z�iNTVU-\�U�a \kZ�PN�#Z�aVU�noTV\�\dnoT�P�Zfq�noPNadnoTVUCWLn q-_badaV_bs�U�a�WLn�q�iNadTV_bs`KNO�U�noP�U�[IU�s4a��4nEU�P��7nEU-\Z�sosEp-U-\�}oPNnE\ akZ�TVW:�LP�_bTV\ W�� iNs4�#Zb\di�_bsE\ WLnY[���[IU~a 7}�UF�LP�U-\�U�s4aV_b[I_bP�_bt#i�\VaVU-\^\VU~[I_bs adTV��TVU�s4a~adTV�-\~Wop�R Z�iNadi�\daVU-\~�7n:Z�s4aJZ�n \VnEq-q-�-\CWLn q-_bs4adT�$bP�U��oi�_bP�_bt#i��7nEU#K _ _bnoTJU�nL�:�:P�U\dnEq-q-�-\gW � ��P�ZYq-_�q-q�iNsEU�PNP�U adi�U�s7a WoUIP�� U��oq-U�}oadi�_bs3�`P�Z0\diNadn:Z�adi�_bs�TVp-U�PNP�U }oTVp-\VU�s4akZ�s7anosEUJq-_b[j}oP�U���iNaVpK�oi�U�s�adTV_b}�iN[j}�_bTdakZ�s4aVUWS���L}o} � �� � M#T2K

]`U-\ R�Zbq�aVU�noTV\^Z�t#i�\V\kZ�s7a \dnoT�P�U~sE_b[K�oTVUgW�� iNsEWLiN�7i�WLnE\QW�� nosEUJ}�_b}onoP�Z�adi�_bs p�akZ�i�U�s4aWLiN�7i�\Vp-\�U�s adTV_bi�\�t#TV_bno}�U-\UZ P�U-\�}oTVp-WEZ�aVU�noTV\�U�a�P�U-\�}:Z�TkZb\diNaVU-\���Wo_bs7aYP�U sE_b[K�oTVUWop�}�U�sEW WLiNTVU-q�aVU�[IU�s4a WoU=P�� U-\d}��-q-U�q-_bsE\di�Wop�TVp-UF��P�U-\�Z�t9U�s7aV\jq�PNiN[ Z�adi��4nEU-\j_bn U�s7�7i� TV_bsosEU�[IU�s4akZ�nL�:�"�7noi�U��oU�TVq-U�s4agP�U�noTJiNs��EnEU�sEq-UI\kZ�sE\JU�s \dnN�oiNTfU�s TVU�aV_bnoT��`U�afU�s)EsW���Z�noadTVU-\ }oTVp-WEZ�aVU�noTV\�Z�nITVp�t#iN[IU Z�PNiN[IU�s7akZ�iNTVU^�#Z�Tdi�p��7noiosEU�Wop�}�U�sEWoU�s7a+}:Zb\+WoU P�� U-\ }��-q-U=q-_bsE\di�Wop�TVp-U#K�]"Z�adTV_bi�\Vi���[IU=q�Z�aVp�t9_bTdi�U sEUITVU�}oTVp-\kU�s4aVU��4nEU=}�U�n�W�� iNs4aVp�T���ajWLn

M%�

Page 18: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

M�v ��������� ������� � ����� � ��� ��� �����

}�_biNs4afWoU ��nEUjWLn q-_bs7adT�$bP�Uc�oi�_bP�_bt#i��7nEU#K�]`U-\fWoU�nL� }oTVU�[ji���TVU-\g\V_bs4afWLiNaVU-\ Wop�}�U�s WEZ�s7aVUgWoU~P�ZjWoU�sE\diNaVp � �ETVU-\d}�U-q�adiN�9U�[IU�s7a 2iNsEWop�}�U�sEWEZ�s4aVU WoU~P�ZjWoU�sE\diNaVp �kK

�~�-\CM�X�����9P�U-\�q-_bsLR&TV_bs7akZ�adi�_bsE\�WoU^q-U�adaVU^adrEp-_bTdi�UCZ�nL� Wo_bsosEp-U-\�U���}�p�TdiN[IU�s4akZ�P�U-\q-_b[j[IU�sEq-��TVU�s7a�K"�C� U-\daQq-U �4nEU])EaJ��r:Z�TdP�U-\[O�K@� PNaV_bs^H M�X�����8M�XFXLM J �@}:Z�TQU��oU�[j}oP�UF�Z��9U-q0WoU-\ Wo_bsosEp-U-\=WoUYP�Z��:Z�i�U WoU �^nEWo\k_bsB\dnoTjP�U-\IP�U�[j[jiNsot9\ U�aIP�U-\jPNi�����TVU-\�K e PU-\V\ Zz4ZfWoU^[I_bs4adTVU�T �4nEU^P�U�noT sE_b[!�oTVU �EnEq�adn:Z�iNa�Z-�9U-q^P�U^q&Lq�P�U^WoU-\+akZbqkrEU-\ \V_bP�Z�iNTVU-\�KO�U-\�adTkZ-�#Z�nL� R noTVU�s4a�W�� nos�t#TkZ�sEW iNs4aVp�T���a�}�_bnoT�P�UCq-_b[j[IU�TVq-UJWoUQP�ZfR _bnoTdTdnoTVUF�oZ�P�_bTV\�7nEUF�o}�_bnoT�PNnoi �oiNP`p�akZ�iNa iN[j}�_bTdakZ�s4aQW���Z��bZ�sEq-U�TCWEZ�sE\�P�Z�adrEp-_bTdi�UfWoU~P�� p��9_bPNnoadi�_bs`K�+PNaV_bs \VU WLi�\dadiNsot#n:Z�iNa WoU-\�Z�noadTVU-\gp-q-_bP�_bt#i�\daVU-\%K/e P+}�U�sE\kZ�iNag�7nEUjP�UjR _#\V\VpIU�s7adTVUp-q-_bP�_bt#i�UjU�afp��9_bPNnoadi�_bs p�akZ�iNaJadTV_b}et#TkZ�sEW�K _ _bnoTJPNnoi ��P�Z=}oTVU�[ji���TVUjU-\da~iN[j}�_bTdakZ�s7aVU}�_bnoTgP�Z�adrEp-_bTdi�U=WoUIP�� p��9_bPNnoadi�_bs U�afiNP �9_bnoP�Z�iNa �4nEUIP�� p�adnEWoU WoU-\ }�_b}onoP�Z�adi�_bsE\ \V_biNa��nEUJq-_b[j[IUf}:Z�Tdadi�UJiNs7aVp�t#TkZ�s4aVU WoUJq-UfWo_b[ Z�iNsEU�S��N�L}o} � �� � �dT2K

e Pg\�� iNs7aVp�TVU-\V\kZ � WoU�nL� }oTV_��oP���[IU-\ U�s }:Z�Tdadi�q�noPNi�U�T�KJ��U�TdaVU-\�� P���Z�}o}:Z�TdiNadi�_bs�WoUakZbqkrEU-\ q-_bP�_bTVp-U-\I\dnoT�P�Z R _bnoTdTdnoTVUYWoU-\ Z�soiN[ Z�nL� U�s t#noi�\VU�WoUYq�Z�[I_bn��@Z�t9U�U-\dajP�UR TdnoiNa+WoU P�� p��9_bPNnoadi�_bs`KY� Z�i�\ q-_b[j[IU�s4a�U���}oPNi��7nEU�T+�4nEU�q-U�\k_biNa�aVU�PNP�U�R _bTd[IU�WoU akZbqkrEU-\}oPNnoa�$ba+�7n`� nosEU�Z�noadTVU��7noio\V_biNa+Z�}o}:Z�TdnEU���]`U�WoU�nL�Li���[IU�}oTV_��oP���[IU�U-\da�P�� _bTdiNt#iNsEU�WoU-\WLi�"�p�TVU�sEq-U-\fU�s4adTVU WoU-\JU-\d}��-q-U-\g�4noi \VU TVU-\V\VU�[!�oP�U�s4a[��U�Z�nEq-_bno} Z�P�U-\Jt#TkZ�sEWoU-\fWLiSR< R p�TVU�sEq-U-\f\V_bs7afWLnEU-\K� P���ZbWEZ�}oakZ�adi�_bs3��[ Z�i�\JP�U-\J}�U�adiNaVU-\~sEU�}�U�no�9U�s4a[��p�sEp#):q�i�U�TgWoUq-U�adaVU � nE\dadi�):q�Z�adi�_bs`K�+PNaV_bs=\VU \ViNadn:Z�iNa �~[ji� 2qVrEU�[jiNs�U�s4adTVUQq-U�nL�I�4noiETVU �dU�akZ�i�U�s4a�P�U q-_bsEq-U�}oa�WoU^\Vp�P�U-q# adi�_bs s:Z�adnoTVU�PNP�U~U�a�q-U�nL� }�_bnoT��7noi�q#� p�akZ�iNa�nosEUQTVp�Z�PNiNaVpCZ��9p�TVp-U#K e P:}�U�sE\kZ�iNa �4nEUQq-U�adaVUWoU�Tdsoi���TVUCsEU~\�� U��oU�T�b�Z�iNa�}:Zb\ WoU^R Z�b-_bs�q-_bsE\dakZ�s4aVU#Ko]"_bTV\V�4nEUQP�UQsE_b[K�oTVUJW�� iNsEWLiN��i�WLnE\U-\da�iN[j}�_bTdakZ�s4a��9U�PNP�U R Z��9_bTdi�\VU�q-U�nL� �7noiLTVp-\di�\daVU�s4a+Z�nL�g[ Z�P�ZbWLi�U-\ _bnIZ�nL� }oTVp-WEZ�aVU�noTV\Z�P�_bTV\~�7nEUgP�_bTV\V�7nEU P�U�noTCsE_b[!�oTVU�U-\daQTVU-\VadTVU�iNs4a���U�PNP�U }�U�noaJZ�PNP�U�T�� nE\V�7n`� �Iq-U-\k\VU�T�K@Oq-UJ[I_b[IU�s7a �PM��:WoU-\ q�Z�TkZbq�aVp�Tdi�\Vadi��4nEU-\Q}�U�no�9U�s7aQZ�}o}:Z�TkZ � adTVUg\ Z�sE\�nos�TVp-U�P:��U-\V_biNs`K

��U�adaVUgq-_bsEq-U�}oadi�_bs Z�}o}�_bTdaVU~nosEU~TVp�}�_bsE\VU~}oP�Z�nE\Vi��oP�UgZ�nL��WoU�nL��}oTV_��oP���[IU-\^�7noiZ��bZ�i�U�s7a~Z�adadiNTVp P���Z�adaVU�s4adi�_bseWLn ��TdiNakZ�sosoi��7nEU#K:�JZ�sE\QP�� p�P�Z���_bTkZ�adi�_bs WoUgq-U�PNP�U# 2q�i ��\V_bs�onoa p�akZ�iNa�WoU~[I_bs4adTVU�T �7nEUCP���Z�s:Z�P��\kUJWLn sE_b[!�oTVUfW�� iNsEWLiN��i�WLnE\ W�� nosEUJU-\d}��-q-UQ}�U�noasE_bnE\jZ�}o}oTVU�sEWLTVU���U�Z�nEq-_bno} \VnoT \V_bs p��9_bPNnoadi�_bs S���"}`K � { � �~iNsot9\dP�Z�sEW TVU�s4�9_bi�U9��I;3. � -049* =>E04YE��� WoUg��r:Z�TdP�U-\ � PNaV_bsT2K

O P�Z [���[IUfp�}�_7�7nEUF��P�U-\ U��L}�p�Tdi�U�sEq-U-\ U�sYP�Z���_bTkZ�aV_biNTVUf\VUJWop��9U�P�_b}o}:Z�i�U�s7a���Z��9U-q}�_bnoT]�onoafWoU TVp-WLnoiNTVUIP�Z=q-_b[j}oP�U��LiNaVpjWoU�P�Z=s:Z�adnoTVUjU�sei�\V_bP�Z�s4aJnosEU }�U�adiNaVU�}:Z�Tdadi�UWoU�q-U�PNP�U# 2q�i�K���_b[j[IU Z�no}:Z�TkZ��bZ�s7a�� P�Ze}oTdiNsEq�iN}:Z�P�U q-_bsE\Vp-�7nEU�sEq-UYR noaIP�Z }oTdi�\kU-\ WoUq-_bsE\Vq�i�U�sEq-U WoU-\ \Vq�i�U�s7adi�):�4nEU-\^P�� p�aVU�sEWLnEUgWoU~P�U�noT iNt#sE_bTkZ�sEq-U#K

8_bnoaVU�R _bi�\�� �Q_�4Z�P �C_bTdaV_bs ��r:Z�}o[ Z�s H M�v#vFX� M�X��FX J ��U�s4aV_b[I_bP�_bt#i�\daVU9�0P�� nosoi� �9U�TV\diNaVp WLn��0iNsosEU-\k_bakZ0Z�n�Wop&�onoagWLn�\di��-q�P�UF�`�7noir)EagWoU-\gp�adnEWoU-\ U�s P�Z���_bTkZ�aV_biNTVU-\

Page 19: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

M�X}�U�sEWEZ�s7a U�a Z�}oTV�-\gP�Z�}oTVU�[ji���TVUIt#nEU�TdTVU [I_bsEWLi�Z�P�UF�`}oTV_b[gnoajZbq�adiN�9U�[IU�s4a P�UIWop��9U# P�_b}o}�U�[IU�s7aQWoUJq-U�PNP�U-\ 2q�i�K

� Z�TVWoU�Tg��P�Y"�OQPNP�U-UF���4n:Z�s7a[�IPNnoi �@TVp�Z�PNi�\kZ=P�� iN[j}�_bTdakZ�sEq-UjW�� p�adnEWLi�U�TfWoU-\~q-_b[I [gnos:Z�noaVp-\f}oPNnoa�$ba �4nEUjWoU-\~iNsEWLiN��i�WLnE\���P�� iNs��EnEU�sEq-UI�7nEUjq-U�nL� 2q�i U��LU�TVq-U�s7afP�U-\~nosE\\dnoT"P�U-\ Z�noadTVU-\�p�akZ�s4a�}oTdiN[I_bTVWLi�Z�P�U#K _ _bnoT�PNnoi �bp-q-_bP�_bt#i�U U�a�\V_7q�i�_bP�_bt#i�U�\V_bs7a8adTV�-\�PNi�p-U-\�K

��U-\�p�P�Z�sE\�p�akZ�i�U�s7a \V_b[j[IU aV_bnoaVUeadTV�-\0WLi�\Vq�TVU�aV\�U�aYsEUe}�U�Td[jiNTVU�s4a�}:Zb\ Z�nL�W7s:Z�[ji��7nEU-\IWoU�}�_b}onoP�Z�adi�_bsBWoU�q-_bsos:Z � adTVU�nos TVp-U�P U-\V\V_bT�K�e P�\�[I_bs4adTV��TVU�s7a=aV_bnoaWoUg[���[IU��7nEUfP�Z WLiNTVU-q�adi�_bs5� \dnoiN��TVU p�akZ�iNaQPNi�p-UjZ�nL��[ Z�adrEp�[ Z�adi��7nEU-\�K"�C� U-\daR� q-U[I_b[IU�s4a �PM�J�7nEU �CZz�[I_bsEW _ U�Z�TdPEU�a \kZCq-_bnoTp��U�P�_bt#i�\dadi��4nEU0�d_bnE��TVU�s4a nos�t#TkZ�sEW�T�$bP�US���L}`K �F� T2K

�QZ�7[I_bsEW _ U�Z�TdP%)Ea+\kZQadrE�-\VU�WoU^Wo_7q�aV_bTkZ�a\�CP�� nosoiN�9U�TV\diNaVp^WLn��0i�qkroiNt4Z�s=\V_bnE\�P�ZWLiNTVU-q�adi�_bs�WoU �CU�Tp��U�Tda O7}�U�sEq-U�T X9U�sosoiNsot9\ H M�v#m#v�LM�X { � J ��P�U-�4nEU�P�P�� iNsoiNadi�Z�iNa��b�4nEU�P��7nEU-\Z�sosEp-U-\�}oPNnE\�a�$ba���fP�Z 2sE_bno�9U�PNP�U(�oi�_bP�_bt#i�U �kK e P:adTkZ��bZ�iNPNP�Z U�sE\dnoiNaVU]�f]"U�iN} �-iNt%� �~Z�}oP�U-\U�a ]`_bsEWLTVU-\��E_�v�iNP`q-_bsos7noa �fZ�TdP _ U�Z�TV\V_bs^H M�v � � �`M�X��#m J K_ U�Z�TdP�Z�}o}oTVp-q�i�ZJ}:Z�Tdadi�q�noPNi���TVU�[IU�s7a � �36 �c2�- � � -025E�� ��=>.<6%;3=>6>�4�7nEU _ U�Z�T \V_bs�Z-�#Z�iNag}onN�oPNi�p=U�sBM�vFX���8Z�P�_bTV\ �4nEU P�U-\[�oi�_bP�_bt#i�\daVU-\ WoUIP�� p�}�_7�7nEUjP���Z��bZ�i�U�s7a�Zbq# q�nEU�iNPNPNi�Z-�9U-qQnosEU q-U�TdakZ�iNsEUQTVU�aVU�s4nEU#K��JZ�sE\�q-U�PNiN��TVUF� _ U�Z�TV\V_bs�U���}�_#\VU�P�Z~R Z�b-_bs Wo_bs4anos0\Vq�i�U�s4adi�):�7nEUgWo_biNaQadTkZ-�#Z�iNPNP�U�T!ZEq�P�Zb\k\di�):U�T^P�U-\ R Z�iNaV\ U�aCWop-q-_bno�7TdiNTCq-Uf�7noi`P�U-\ TVU�PNi�UU�a U�s)Es0U���}oTdiN[IU�TQ\VU-\�TVp-\dnoPNakZ�aV\Q\V_bnE\ P�Z R _bTd[IUJP�Z }oPNnE\ p-q-_bsE_b[ji��7nEUg�7noi`\V_biNa[Z

rEU \diNsot#P�U \VakZ�aVU�[IU�s4a���adrEUK�oTdi�U�R R _bTd[gnoP�Z��adrEUgR U � ��_bTVWo\J_�R ��roi�qkrTVU�}oP�Zbq-UCiNs _bnoT�[jiNsEWo\�Z���i�WoUCTkZ�sot9U~_�R`TVU�P�Z�adi�_bsE\droiN}E\ ��U�a ��U-U�sYi�\V_bP�Zd aVU-WY}orEU�sE_b[IU�s:Z�@i�\ ��r:Z�a ��UfaVU�Td[�Zj\kq�i�U�s4adi�):qfP�Z � S��N�L}`K � v� �~iNsot� \dP�Z�sEWYq�iNaVU � �36�I2�- � �L-/2 E�� ��=>.<6%;3=>6YWoU _ U�Z�TV\V_bsT2K

]`UCadTkZ-�#Z�iNP`WLn�\Vq�i�U�s4adi�):�7nEUJq-_bs4adiNs7nEUfnosEUQR _bi�\ P�Z P�_bi�R _bTd[gnoP�p-U�Z�iNP�Wo_biNa \kZ�sE\�q-U-\V\VUP�Zft9p�sEp�TkZ�PNi�\VU�T U�a�P���ZdPIsEU�T�}�_bnoT��7n`� U�PNP�U~q-_bTdTVU-\d}�_bsEWoUJZ�n�[ji�U�nL� � P�ZgTVp�Z�PNiNaVpJWoU^P�Zs:Z�adnoTVU#K_ _bnoT _ U�Z�TV\V_bs3�Lq-U�adaVUCZ�}o}oTV_�qVrEU~\Vq�i�U�s7adi�):�4nEU~p�akZ�iNa nosEUQ}oroiNP�_#\V_b}oroi�U~WoU^P�Zf�7i�U#Ke2P }�U�sE\kZ�iNag�7nEUIq#� U-\da Z�iNsE\Vi �4n`� iNP+R Z�noag\kUIq-_b[j}�_bTdaVU�T�Z�n��7nE_badi�WLi�U�s3�"�7nEUIq#� U-\dafP�Zq-_bsEWLiNadi�_bs \diNsEU �4n:ZCsE_bsIWLnI\ Z-�9_biNT+U�a+�7n`� U�PNP�U�\VU�noP�U }�U�Td[IU�a\�CP�ZC\V_�q�i�p�aVp�W�� p��9_bPNnEU�TS���L}o} � md � vdT2K

_ _bnoT=nE\VU�TIWoU�q-U�adaVU0Z�}o}oTV_�qVrEU U�s �oi�_bP�_bt#i�UF�+iNP R�Z�PNP�Z�iNaIaV_bnoa W���Z���_bTVW Wop#)EsoiNTq�P�Z�iNTVU�[IU�s4a P�U-\8aVU�Td[IU-\ q-U�s4adTkZ�nL�:��q-_b[j[IU \Vp�P�U-q�adi�_bs � _bn�2rEp�TVp-WLiNaVp � �#�7noi4s`� p�akZ�i�U�s7a� nE\V�7nEU# �PM� �7nEU+adTV�-\8�#Z�t#nEU-\�K _ U�Z�TV\V_bs�\VUr)o�EZ^}�_bnoT/�onoa�WoU [I_bs4adTVU�T �4nEU�P�� p��9_bPNnoadi�_bsU-\da�P�Z�q-_bsE\Vp-�7nEU�sEq-UfWoU~R�Zbq�aVU�noTV\ [IU-\dnoTkZ��oP�U-\�K

]`_bTV\jWoU�\V_bs TVU�aV_bnoT Z�n }:ZzL\�� _ U�Z�TdPQZ��bZ�iNaIU�s a���aVU�WoU�q-_b[j[gnosoi��7nEU�T=q-U�adaVU�7i�\di�_bs WoU-\ qVrE_#\VU-\L� \VU-\�q-_bPNP���t#nEU-\ Z�[Ip�Tdi�q�Z�iNsE\�Ke2P�Zd"�U-q�adi�_bsos:Z�iNa }:Z�Tdadi�q�noPNi���TVU�[IU�s7a

Page 20: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

��� ��������� ������� � ����� � ��� ��� �����

P�Zf�7i�\di�_bs \dakZ�adi�\dadi��7nEUQWoU _ U�Z�TV\k_bs`Ko��U�}�U�sEWEZ�s4a���P�U�noTV\�}�_biNs7aV\ WoU ��nEU^WLiN�9U�Tdt9U�Z�i�U�s4a\dnoT�nosjWop�akZ�iNP:Z P�Z^adrEp-_bTdi�U WLnjq-p�P��&�oTVU ��_bakZ�soi�\daVU�Z�noadTdi�qVroi�U�s � U�sEWoU�P3H M�v�����LM�v#v {7J �Wo_bs7a^iNP"p�akZ�iNaQ}:Z�Tdadi�\kZ�s3�E[ Z�i�\Q�4nEU _ U�Z�TV\V_bs TVU � U�akZ�iNa�K@|^sEUfU��L}�p�Tdi�U�sEq-Ug�7n`� iNP`[IU�s:Z}:Z�TkZ�PNP���P�U�[IU�s7aJZ��9U-qgX9U�sosoiNsot9\Q[ Z�i�\^WoUf[ Z�soi���TVUfiNsEWop�}�U�sEWEZ�s4aVU [jiNaQU�s0p���i�WoU�sEq-UP���Z��E\VU�sEq-UJWoUCPNi�U�sYU�s4adTVU~P�U-\�EnEq�adn:Z�adi�_bsE\^W�� nosEUJU-\d}��-q-U~U�a�\V_bsYp��9_bPNnoadi�_bs`K:��U�P�Z P�Uq-_bsLR _bTdakZjWEZ�sE\^\k_bs0Zbq-q-U�}oakZ�adi�_bs WoUJP�Z adrEp-_bTdi�UfWoUK� U�sEWoU�P�K�+sBM�XLM�m��P�Z TVp�}onoakZ�adi�_bs�WoU _ U�Z�TdP�U-\daJaVU�PNP�UI�7n`� iNP+U-\dafp�PNn � P�� O�q�ZbWop�[ji�U �~Zd adi�_bs:Z�P�UYWoU-\�O�q�i�U�sEq-U-\�Kr� s M�XLM�v��iNP�U-\da sE_b[j[Ip�}oTV_�R U-\V\kU�noT�WoU �oi�_b[Ip�adTdi�UYU�ajWoU\dakZ�adi�\Vadi��4nEU-\ WLn���iN�bZ�s7a���P�� nosoiN�9U�TV\diNaVp X9_brosE\ �Q_b}oc�iNsE\��`_�v X9U�sosoiNsot9\ adTkZ-�#Z�iNPNP�Z�iNaWop ���LK:|QsEUgZ�sosEp-U }oPNnE\^akZ�TVW:�:nos0iNsEq-U�sEWLi�UfTkZ��bZ�t9U P�U[� � adiN[IU�s7aC_�v iNP8U-\da^iNsE\dakZ�PNP�pF�q�Z�nE\kZ�s7a P�Zf}�U�TdaVU WoU aV_bnE\ P�U-\�Wo_7q�no[IU�s7aV\�U�a Wo_bsosEp-U-\��7n`� iNP@Z-�#Z�iNa�Zbq-q�no[ noP�p-\�}�U�s WEZ�s7a nosEU0�7iNsot#akZ�iNsEU W���Z�sosEp-U-\�K�]+� p�adnEWoU \dnoT=P�Z adnN��U�TVq�noP�_#\VU �4n`� iNPCp�akZ�iNa \dnoT P�U}�_biNs4a^W���ZbqkrEU��9U�TQU-\da��:U�s4adTVUgZ�noadTVU-\��ETVp-WLnoiNaVUK��sEp�Z�s4a�K

O �:Z�adadn�}:Z�T�q-UCq-_bno}�WLn \V_bTda�� _ U�Z�TdP�TVU�sE_bsEq-U]�gq-U�adaVUCTVU-qVrEU�TVqVrEUJU�a \VUQ}�U�sEqkrEU\dnoTJP�U�}oTV_��oP���[IU WoU�P�Z�\dnoTd}�_b}onoP�Z�adi�_bs`K"O q-U�adaVUI)Es3��iNP �9U�noafp�adnEWLi�U�TgWoU-\g\V_bnoTdi�\��[ Z�i�\�P�Zfq-_bP�_bsoi�U~�7n`� iNP�Z��bZ�iNa�q-_b[j[IU�sEq-p-UJZJ}�p�Tdi�WEZ�sE\�P�� iNsEq-U�sEWLi�U#K:�C� U-\da�}�_bnoTV�7nE_bi �U�s�Wop-q-U�[!�oTVU M�XLM�X�oiNP`q-_b[j[IU�sEUgnos�p�P�U��bZ�t9UfWoUJWLTV_#\k_b}oroiNP�U-\��E�7noi`_bs4a P���Z��bZ�s7akZ�t9UW���Z��9_biNTQWop ������U�Z�nEq-_bno} p�aVpJp�adnEWLi�p-U-\KS��N�L}o} � X�:m��dT2K

_ _bnoT _ U�Z�TdP �`P�U-\g\dakZ�adi�\dadi��7nEU-\fWLn ��iN�bZ�s7aWH&akZ�nL�eWoU�s:Z�akZ�PNiNaVpF�8WoUjWop-q-�-\��`U-\d}�p# TkZ�sEq-U=WoUI��i�UF�NKNKNK J \V_bs7agiNsEWLi�\d}�U�sE\kZ��oP�U-\I�YP���Z-�#Z�sEq-p-U WoUIP�Z��oi�_bP�_bt#i�U U�a���P�Z0q-_b[I }oTVp�rEU�sE\di�_bs WoU P�� p��9_bPNnoadi�_bs`K _ _bnoTfp-q�P�Z�iNTVq�iNTCP�U-\CPNi�U�sE\JU�s7adTVU q-U�adaVU�WoU�Tdsoi���TVUjU�a~q-U-\\dakZ�adi�\Vadi��4nEU-\���iNP��9U�noaIU��L}�p�TdiN[IU�s4aVU�T P�U-\�[Ip�adrE_�WoU-\IWoU�P�Z Wop�[I_bt#TkZ�}oroi�U \dnoTIWoU-\}�_b}onoP�Z�adi�_bsE\ Z�soiN[ Z�P�U-\�K e2P@[I_bs4adTkZ �4nEUQP�U-\�q-_bnoTp��U-\�Wop-q�TdiNaVU-\ }:Z�T�q-U-\�}�_b}onoP�Z�adi�_bsE\\V_bs7a^U�s�sE_b[!�oTVUJTVp-WLnoiNa�K_ U�Z�TdP"�9_�4Z�iNaQP�U-\^qVrE_#\VU-\QW�� nosEUJR�Z�b-_bsY}:Z�Tdadi�q�noPNi���TVU#Kue P"Z�PNP�Z�iNa � nE\V�7n`� ��q-_bsE\di�Wop# TVU�TfP�� U-\V}�p�TkZ�sEq-UIWoU���i�Ujq-_b[j[IUjP�� U��L}oTVU-\V\di�_bs�WoU�P�� _bTdt4Z�soi�\ Z�adi�_bs R _bsEq�adi�_bsosEU�PNP�U U�a\dadTdnEq�adnoTVU�PNP�U WoU-\�iNsEWLiN��i�WLnE\^W�� nosEUgU-\V}��-q-UJU�a sE_bs0}:Zb\ q-_b[j[IUfnosEUf\ViN[j}oP�Ufq�Z�TkZbq# aVp�Tdi�\dadi��7nEU �oi�_bP�_bt#i��4nEU#Ko��U�adaVUQq-_bsEq-U�}oadi�_bs�TkZbWLi�q�Z�P�UCWoU-\�qkrE_#\VU-\ R&noa�}�U�noa a��adTVUF�7nosEUR _bi�\�P�� rEU�noTVU WoU-\ q�TdiNadi��7nEU-\ �9U�s4nEUF�bP�ZCTkZ�i�\V_bsIWoU�q-U�TdakZ�iNsEU-\ R&Tdi�q�adi�_bsE\ Z��9U-q�q-U�nL��Wo_bs7aP�UJ}�_biNs4aQWoU~�7nEUJp�akZ�iNa^WLi�"�p�TVU�s4a�K

]"UIadrE��[IU �7nEUjadTkZ�iNaVU _ U�Z�TdP�aV_bnoa Z�n P�_bsot WoUI\V_bs adTkZ��bZ�iNP�q-_bsE\di�\daVU �YWop�aVU�T [jiNsEU�T�\di nosEU q�Z�TkZbq�aVp�Tdi�\dadi��4nEU U-\da=P�UYR TdnoiNa=WoU�P�� rEp�TVp-WLiNaVp _bn3��Z�n q-_bs4adTkZ�iNTVUF� WoUP�� U�s7�7iNTV_bsosEU�[IU�s7a�K"O q-U�adaVUK)Es3��iNP8noadiNPNi�\VU nosEU [Ip�adrE_�WoU \kq�i�U�s4adi�):�7nEUjU�s q-_bsLR TV_bs akZ�s7a~WoU-\Q[IU-\dnoTVU-\QTVp-\dnoPNakZ�s4aV\JWoU WLi�"�p�TVU�s7aV\CR Zbq�aVU�noTV\��:q#� U-\da �d 2WLiNTVU�WoU q-_bsEWLiNadi�_bsE\U��L}�p�TdiN[IU�s7akZ�P�U-\QWLi�"�p�TVU�s4aVU-\!S��N�L}o}0m���:mLM#T2K

OQP�_bTV\��4n`� iNP�Z-�#Z�iNaj�4n:Z�TkZ�s7aVU# 2�4n:Z�adTVU�Z�sE\�U�a��4n`� iNP p�akZ�iNa���P�� nosoiN�9U�TV\ViNaVp X4_brosE\�C_b}oc7iNsE\JWoU�}onoi�\gq�iNsE� Z�sE\�� _ U�Z�TdP \VU ��iNa~_�"�TdiNT~P�� _b}o}�_bTdadnosoiNaVp WoU�adTkZ-�#Z�iNPNP�U�TgWEZ�sE\

Page 21: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

�LM

P�� iNsEWLnE\dadTdi�U~}�_bnoT nosEU~q-_b[j}:Z�t#soi�UJsEU � 9_bTdc#Z�i�\VU#KoOQ}oTV�-\ }oPNnE\di�U�noTV\ [I_bi�\�W�� rEp-\diNakZd adi�_bs3�`iNP TVU�sE_bs%b�Z�`[I_badiN�9p=U�s4adTVU�Z�noadTVU-\g}:Z�TgP�U-\fsE_bno�9U�PNP�U-\ q-_bsEWLiNadi�_bsE\ }�_bnoTfP�U[I_biNsE\fZ�PNP�p-qVr:Z�s4aVU-\ �7nEU�PNnoi�}oTV_b}�_#\kZ P�� nosoiN�9U�TV\diNaVpBZ@nosEU�}oPNnE\~t#TkZ�sEWoUIPNi���U�TdaVpF��nosadiNadTVUfWoU _ TV_�R U-\V\VU�noT^WoU]�oi�_bP�_bt#i�U U�a�P�Z�TVp-WLnEq�adi�_bs WoUJ\kZ�qkr:Z�Tdt9UgW�� U�sE\VU�iNt#sEU�[IU�s7a�K_ _bnoT [IU�sEU�T ���oi�U�s nos }oTV_��dU�a WoU�t#TkZ�sEWoU�U�s7�9U�Tdt#noTVU \dnoTjP�Z WLnoTVp-U0WoU��7i�UQ [IU�adadTVUfU�s�p��7i�WoU�sEq-UfP�� iN[j}:Zbq�a^WoU~R�Zbq�aVU�noTV\ q-_b[j[IUJP�Z�WoU�sE\diNaVpfWoU~}�_b}onoP�Z�adi�_bs3�P�� rEp�TVp-WLiNaVpF�4U�aVq#KNKNK�QuiNPEWoU�[ Z�sEWEZfnos��onEWLt9U�a W�� nosI[jiNPNPNi�_bs WoU Wo_bPNP�Z�TV\��9q-U��7noi:p�akZ�iNap�sE_bTd[IU#K�e PQWLnoaY\VU q-_bs4aVU�s7aVU�T0W�� nos [I_bs7akZ�s4a�}oPNnE\ [I_�WoU-\daVUF���4noiCPNnoiC\kU�[K�oP�Z�iNaiNsE\dnP \kZ�s4a�KO�_bs�P�Z���_bTkZ�aV_biNTVUYp�akZ�iNa nosEU=nosoiNaVp=iNsEWop�}�U�sEWEZ�s4aVUYWEZ�sE\�P�� nosoiN�9U�TV\diNaVp�U�a���Z-�9U-q�4noiNs ��UQ[jiNPNP�U~Wo_bPNP�Z�TV\ }:Z�T�Z�sE\��4iNP�p�akZ�iNa�}:Z�Td[ji�P�U-\�}oTV_�R U-\V\VU�noTV\�P�U-\�[ji�U�nL� }:Z�9p-\�KNe2Pp�akZ�iNaCWoU }oPNnE\^}oTVU-\k�4nEU aV_bakZ�P�U�[IU�s4a~PNi��oTVU�WoUfaV_bnoaVU a � qVrEU W�� U�sE\VU�iNt#sEU�[IU�s7a�S��N�E}o}m���Em��dT2K

�JZ�sE\ P�UCq-_bs4aVU��LaVUQWoUQP�Z _ TVU�[ji���TVU �CnEU�TdTVU � _bsEWLi�Z�P�UF�7iNP�p�akZ�iNa q-_b[j[gnosEp�[IU�s7aZbWL[ji�\ �7nEUIt#nEU�TdTVU�U�a [I_�9U�sE\jWoU \dnN�E\di�\dakZ�sEq-U p�akZ�i�U�s4ajp�adTV_biNaVU�[IU�s4ajPNi�p-\�U�a �4nEUP�U-\f\dakZ�adi�\dadi��7nEU-\gp�akZ�i�U�s7agP�� _bnoadiNP i�Wop�Z�P�}�_bnoT p�adnEWLi�U�T q-U�adaVUjTVU�P�Z�adi�_bs`K8O^}oTV�-\ Z-�9_biNT[ji�\^U�s0p���i�WoU�sEq-UJP�� iN[j}:Zbq�a~�4nEU~}�U�noaQZ-�9_biNTCnos0q-_bs��EiNa��:}:Z�T P�� iNs4aVU�Td[Ip-WLi�Z�iNTVU�WoUJP�U}�p�s7noTdi�U Z�PNiN[IU�s4akZ�iNTVUF�4\VnoT�nosEU }�_b}onoP�Z�adi�_bs3� _ U�Z�TdP:\���Z�adaVU�P�Z~\Vp�Tdi�U�nE\kU�[IU�s4a �QP�� p�adnEWoUWoUJq-U-\ WoU�Tdsoi���TVU-\�K� s M�X�����7iNP�adTkZ-�#Z�iNPNP�Z�iNa�\dnoT�P�U-\ akZ�nL�=WoUCq�TV_bi�\V\kZ�sEq-U~WoU-\�}�_b}onoP�Z�adi�_bsE\(Z7iNP�qVrEU�T qVr:Z�iNa��JP�U-\ U��L}oTdiN[IU�T [ Z�adrEp�[ Z�adi��7nEU�[IU�s4a��N�~P�U-\+[IU�adadTVUQU�s=p-�7n:Z�adi�_bsE\�KYe Po�9_bnoP�Z�iNap�akZ��oPNiNT+nosEU�P�_bi:�7noio}oTVp-WLi�\VU�nosEU�aVU�sEWEZ�sEq-U TkZ�i�\V_bsos:Z��oP�U^U�a �CP�Zb�7nEU�PNP�U \VU�q-_bsLR _bTd[IUP�UJq-_b[j}�_bTdaVU�[IU�s7aCWoU-\�}�_b}onoP�Z�adi�_bsE\�K

]`U�TVp-\VnoPNakZ�a�Z�nE�7nEU�PQiNP~Z���_bnoadiNa U-\da nosEU p-�7n:Z�adi�_bs U�sBR _bTd[IU WoU�O aVU�sEWEZ�s4a�9U�TV\ nosEUJPNiN[jiNaVU \dno}�p�Tdi�U�noTVU#K���U�adaVUgp-�7n:Z�adi�_bs0\VUJ�9U�TdTkZI}oPNnE\ akZ�TVW Wo_bsosEU�T^P�UJsE_b[W��Mwpx&yN{F|~}~�F� ������}?�#|~}�x&y%��K"��Uj�4nEU _ U�Z�TdP+iNt#sE_bTkZ�iNa���q#� U-\daJ�4nEU r7noiNakZ�s4aVU Z�sE\~}oPNnE\~a�$bah+U�Tdr4noP�\da�noadiNPNi�\kZ�iNa�q-U�adaVU�p-�4n:Z�adi�_bs }�_bnoT�Wop-q�TdiNTVU P�� p��9_bPNnoadi�_bs W�� nosEU=}�_b}onoP�Z�adi�_bs`K]+� i�Wop-U��9U�s:Z�iNagU�s R�Z�iNafWoUWVQnEp�aVU�P�U�a��`�4noi�}�U�sE\kZ�iNaf�7nEU�P�Z TVp-\di�\dakZ�sEq-UIZ�n Wop��9U�P�_b} }�U�[IU�s7a�WoU��#Z�iNac��adTVU }oTV_b}�_bTdadi�_bsosEU�PNP�U0Z�n q�Z�TdTVp�WoU P�Z �7iNaVU-\V\kU=WoU�q�TV_bi�\V\kZ�sEq-U�WoUP�Z }�_b}onoP�Z�adi�_bs3��}:Z�T�Z�s:Z�P�_bt#i�UU��P�ZeR _bTVq-U WoU0R TV_badaVU�[IU�s4aYU�s }or7�\Vi��4nEUF� �7noiQU-\da}oTV_b}�_bTdadi�_bsosEU�PNP�U Z�n0q�Z�TdTVpfWoU~P�Z���iNaVU-\V\VU#K

]"Z [���[IUfq-_bnoTp��Uf\diNt#[I_���WoU~R noa�noadiNPNi�\Vp-UJU�s�M�X��#v }:Z�T� JK ��TkZ�iNP�\dR _bTVW �Q_���U�Tda \V_bs^H M�v#v { ��M�X���� J Zd)EseWoU Wop-q�TdiNTVU P�U Wop��9U�P�_b}o}�U�[IU�s7agWoUg}oP�Z�s7aVU-\���W���Z�soiN[ Z�nL� U�aW�����adTVU-\�r4no[ Z�iNsE\�KNe P�PNnoi`Wo_bsos:Z P�U~sE_b[ W��Mwpx&yN{F|~}����5{�y | ����{F| {����F|~}~x&y%� }:Z�T^Z�s:Z�P�_bt#i�U�=WoU-\fTVp�Zbq�adi�_bsE\fqVroiN[ji��7nEU-\g�7noi \���Zbq-q-p�P���TVU�s4agW�� U�PNP�U-\p �[���[IU-\���[���[IUI\di q-U aVU�Td[IUsEU WoU��7TkZ�iNa�\���Z�}o}oPNi��4nEU�T��4n`� �JP�Z~}:Z�Tdadi�U q-U�s4adTkZ�P�UCWoU�P�ZJq-_bnoTp��U[S��N�9}o}=m { m#m� �~iNsot� \dP�Z�sEWYq�iNaVUfU�s4adTVU Z�noadTVU-\ � �36��[- G:65E��� .�0.?;�� WoU _ U�Z�TdP�T2K

Page 22: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

��� ��������� ������� � ����� � ��� ��� �����

Page 23: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� ����������� �

� � �� � � � � �" �� �����

����� � ���������������������! #"$�O^PSR TVU-W X4Z�[IU-\ ]`_badcbZ }onN�oPNi�Zc* 476 � 6%;:G:,IE�� � ��0,d.?=u-/4 ].<E/47E����U�seM�X�� � K

��U�PNiN�7TVU adTkZ�iNaVU WoU-\�}oTV_7q-U-\k\dnE\`W�s:Z�[ji��4nEU-\�s:Z�adnoTVU�P�\���\dn��dU�a`�4noi4Zbq-q�Z�}:Z�TkZ�iNa�O�WEZ�[I\��U�aCU-\da~U�[j}oTVU�iNs4afWoU-\~[Ip�adrE_�WoU-\~WoU P�ZI}or7�\di��7nEU#K>e2P Z�}o}�_bTdaVU sEp�Z�so[I_biNsE\~nosEU iNs sE_��#Z�adi�_bs^Z@\VU�P�_bs ]`_badcbZ��}�_bnoTQq-_b[j}oTVU�sEWLTVU�P�UgR _bsEq�adi�_bsosEU�[IU�s7afW�� nosEUg}:Z�Tdadi�U�WoUP�Z s:Z�adnoTVUF�oiNP"U-\da�iNsEWLi�\d}�U�sE\kZ��oP�UfW�� U�s q-U�TdsEU�T P�Z�t#P�_��:Z�PNiNaVp#K

]"Z!�oi�_bP�_bt#i�U~}orYL\di��7nEUJ\VUQWLi�\VadiNsot#nEUJWoUQP�Z!�oi�_b}or7�\di��7nEUcZoZ�P�_bTV\��4nEU~q-U�adaVUJWoU�T soi���TVU�\VUYq-_bsEq-U�s4adTVU \dnoTjP�U-\I_bTdt4Z�soi�\d[IU-\��+P�Z }oTVU�[ji���TVU Z }�_bnoT \dn��dU�aIW�� p�adnEWoUYP�U[I_bsEWoUfWEZ�sE\ \kZ t#P�_��:Z�PNiNaVp#K@]`_badcbZI\dnoiN�bZ�iNa�PM�jnosY}oTV_7q-p-WopgZ�s:Z�P��adi��7nEUJ}oTV_b}oTVU!� P�ZqVroiN[ji�UC}orYL\di��7nEUF��q-U�TdakZ�iNsEU-\ WoU~\VU-\�a���aVU-\ WoUQqkr:Z�}oiNadTVU-\�Q�q�iNsEp�adi��7nEUF�L\dakZ�adi��7nEUQ_bnU�sEq-_bTVUfW7s:Z�[ji��7nEUfWoU-\ \��\daV��[IU-\ �7iN�#Z�s4aV\�Q U�s�\V_bs4a P�� iNPNPNnE\VadTkZ�adi�_bs S � T2KO�_bs adTkZ��bZ�iNPJU-\da sEp�Z�so[I_biNsE\�adTV�-\ aV_bnoTdsEp �9U�TV\ P�� p-q-_bP�_bt#i�U#K e PQadTkZ�iNaVU sE_bakZ�[I [IU�s4a WoU P�Z+�oi�_bP�_bt#i�U�Zb�4n:Z�adi��7nEUF�"WoU-\ TVp-\VU�Z�nL��Z�PNiN[IU�s4akZ�iNTVU-\���WLn q&�q�P�U=WoUjP�� U�Z�n3�WLn WLi�_-�Li�WoU^WoU q�Z�Tp��_bsEU^U�a�WoU�P���Z ��_baVUF�9adrE��[IU-\�iNPNPNnE\dadTVp-\ WoUL��U�Z�nEq-_bno} WoU^Wo_bsosEp-U-\�4n:Z�s7adiNakZ�adiN�9U-\�K+]+� iN[j}�_bTdakZ�sEq-U�WoU P�� p�adnEWoU�WoU-\jp-qVr:Z�sot9U-\=p�sEU�Tdt9p�adi��4nEU-\ U�s4adTVU�P�U-\_bTdt4Z�soi�\d[IU-\gU-\da \V_bnoPNiNt#sEp-U#K/�+s)Es3�"]`_badc#ZYp-q�P�Z�iNTVU W�� nosEU TdiNt#nEU�noTg[ Z�adrEp�[ Z�adi��7nEUsE_bno�9U�PNP�UjP�U-\J}�_biNs4aV\g�7n`� O�WEZ�[I\fs`��Z��bZ�iNa }one}oTVp-q�i�\VU�T�K`]`U-\I* 476 � 6%;:G:,jTVU-\VaVU�TV_bs4a}�U�sEWEZ�s7a^P�_bsot#aVU�[j}E\CnosEUJTVp�R p�TVU�sEq-UgU�s p-q-_bP�_bt#i�UF�:t#T � q-UfsE_bakZ�[j[IU�s7aR�jP�ZI\VU-q�adi�_bsWEZ�sE\�P�Zb�4nEU�PNP�UfP�U-\ _#\Vq�iNPNP�Z�adi�_bsE\ }oTV_bi�U-\p �}oTVp-WEZ�aVU�noTV\Q\V_bs7a }oTVp-\VU�s7aVp-U-\!S���L}`K[� � T2K

O^P�_bTV\��7n`� iNP s`��Z-�#Z�iNa � nE\V�7nEU# �PM� U�s4adTVU�aVU�s4n Z�nEq�nosEU=TVU�P�Z�adi�_bsBZ��9U-q�P�Z q-_b[j[gn s:Z�noaVp p-q-_bP�_bt#i�\daVUF��]`_badc#Z TVU � _biNt#soiNa~U�s M�X�� � P����+q-_bP�_bt#i�q�Z�P O�_�q�i�U�aa0_�R OQ[IU�Tdi�q�ZLK>e2PU-\da�iNs4aVp�TVU-\V\ Z�s4a~WoUJTVU�[ Z�TV�4nEU�TC�4nEUJ]"_badcbZI�9_bnoP�Z�iNaQq�TVp-U�T nosEUJsE_bno�9U�PNP�U WLi�\Vq�iN}oPNiNsEU\Vq�i�U�s4adi�):�7nEUF�o\kZ�sE\�Z-�9_biNT p�aVpCiNs��EnEU�sEq-p~}:Z�T P�UC[I_bsEWoU]�oi�_bP�_bt#i��4nEUF�oU�a��7nEUQq-U~Wop-\diNTq-_bsEq-_bTVWEZ�iNa WoUI[ Z�soi���TVU �@Z�t#TkZ�s4aVU�Z-�9U-q P�UIWop��9U�P�_b}o}�U�[IU�s7a�s:Z�i�\V\kZ�s4a WoUjP�� p-q-_bP�_� t#i�U#K@��U�P�Z�[I_bs4adTVU�� P�Z R _bi�\ P�UJt9p�soi�UfWLn0\Vq�i�U�s7adi�):�4nEU Z�[Ip�Tdi�q�Z�iNs3�E[ Z�i�\ p�t4Z�P�U�[IU�s7a�4nEU P�UYWop��9U�P�_b}o}�U�[IU�s4a=WoU�q-UYWo_b[ Z�iNsEU�\%� iNsE\Vq�TdiN�bZ�iNajWEZ�sE\jP�Z \VnoiNaVU=P�_bt#i��7nEUYWoU-\p��9p�sEU�[IU�s4aV\�K

���

Page 24: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� { ��������� ������� � � � ��� � ����� � � ���������

] ��Z�s:Z�P��\VU�[ Z�adrEp�[ Z�adi��7nEU WoU�P�� iNs4aVU�TkZbq�adi�_bs U�s7adTVU }�_b}onoP�Z�adi�_bsE\+Z�noTkZQnos�t#TkZ�sEWp-qkrE_~Z�n�adTkZ��9U�TV\ WoU�hCiNaV_Qh+_bPNaVU�TdTkZ�b[ Z�adrEp�[ Z�adi�q�i�U�sjiNakZ�PNi�U�s�WoU�TVU�sE_b[9�#Z-�9U-q�P�U-�7nEU�P]`_badc#Z q-_bPNP�Z���_bTkZLK%]8Z�adrEp-_bTdi�U �7nEU+Wop��9U�P�_b}o}�U�]"_badcbZ�WEZ�sE\�\VU-\/* 4Y6 � 6%;3G>,�Z�s7adi�q�iN}:ZP�Z�adrEp-_bTdi�U WoU-\ \��\daV��[IU-\=aVU�PNP�U �4nEU ]�nEW ��iNt �9_bs � U�TdakZ�P�Z�s"� H M�X��LMF�~M�X9�G� J P�ZWop��9U�P�_b}o}:Z=Z�}oTV�-\gM�X � �LK3S�mdT2K

] � iN[j}:Zbq�a=WLn adTkZ��bZ�iNPCWoU�]`_badc#Z R&noaj}:Z�Tdadi�q�noPNi�U�T Z iNP�}oTV_b}�_#\kZ nosEUY[Ip�adrE_�WoU}�_bnoTfZ�}o}oPNi��4nEU�TfP�U-\~[ Z�adrEp�[ Z�adi��7nEU-\K�=P�Z �oi�_bP�_bt#i�UF��[ Z�i�\f\VU-\Ci�Wop-U-\Jp�akZ�i�U�s7afadTV_b}s:Z � �9U-\�WLn }�_biNs4a�WoU ��nEUD�oi�_bP�_bt#i��7nEU }�_bnoT���adTVU [ji�\VU-\YU�s }oTkZ�adi��7nEUF� [���[IUe\diP�U-\ �4nEU-\dadi�_bsE\g}�_#\kp-U-\gp�akZ�i�U�s4a P�U-\[��_bsosEU-\�K"]"U-\�* 476 � 6%;:G:,=p�akZ�i�U�s4a U�s R�Z�iNafadTV�-\TVp�adTV_#\d}�U-q�adiSR \��oTVU�[IU�adakZ�s4aJZ�n�t9_��oa^WLn�� _bnoT P�U-\ i�Wop-U-\^WLn9']ea'~UJ\di��-q�P�U#K%e P�\ p�akZ�i�U�s7aWoU }oPNnE\C[ Z�TV�4nEp-\C}:Z�TCP�ZI��i�\di�_bs W�� nos Z�[ Z�aVU�noT�K��JZ�sE\~q-U \VU�sE\��:q-UgR&noaQnos p-qkrEU-qF�[ Z�i�\ nosIp-qVrEU-q^�7noiE\dadiN[ noP�ZJP���Z-�#Z�sEq-p-U^WoU P�ZJTVU-qVrEU�TVqVrEUCU�a+�7noioTVU-\dakZ~nosEU�TVp�R p�TVU�sEq-U}�U�sEWEZ�s4aCWoU-\ Wop-q-U�sosoi�U-\KS���o}o}0�#m�¤�9��T2K

e s4aVp�TVU-\k\V_bsE\�sE_bnE\jZ�nL�B�dU�nosEU-\�Z�sosEp-U-\jWoU ]`_badc#Z��q-U�PNP�U-\j�4noi _bs4a�}oTVp-q-p-Wop�P�ZTVp-WEZbq�adi�_bs WoU-\(* 4Y6 � 6%;:G:,9K �QpgWoUJ}:Z�TVU�s4aV\~Z�[Ip�Tdi�q�Z�iNsE\R�j]`U�[K��U�Tdt=U�s O^noadTdi�qkrEUQ ��iNPNP�UIZ�n��d_bnoTVW�� r4noi noc�TkZ�iNsoi�U�sosEU�� U�s M�v#v����iNP R noafp�P�U��9pIU�s�OQPNP�U�[ Z�t#sEUIU�agU�s`oTkZ�sEq-U�Z-�#Z�s4a�W�� _��oaVU�soiNTYnosEU PNi�q-U�sEq-UeU�s }orYL\di��7nEU qVroiN[ji�U�� ��iNTd[jiNsot#r:Z�[ U�sOQsot#P�U�aVU�TdTVU#Ke Pop�adnEWLi�U�U�sE\dnoiNaVU nosEU�Z�sosEp-UL�C]`U�iN} �-iNt%�#}>$bP�U�\kq�i�U�s4adi�):�7nEU WoU�}orYL\di��4nEU qkroiN[ji�U�CP�� p�}�_7�7nEUF�#_�vjiNPE\dnN�oiNa P�� iNs��EnEU�sEq-UQWoU `oTdi�U-WLTdi�qVr � iNPNrEU�PN[ �C\da ��Z�P�W H M�v � ��EM�X���� J �\VU�P�_bs �4noi`P�� p�sEU�Tdt#i�U U-\da nos0q-_bsEq-U�}oa^R _bsEWEZ�[IU�s7akZ�P�Z�nE\V\Vi3�oi�U�s0U�s�}or7�\di��7nEUg�7n`� U�s�oi�_bP�_bt#i�U#K4��U�R noa�q-U�adaVU�i�Wop-U��7noi �bU�s4adTVU Z�noadTVU-\���[IU�s:ZC]`_badc#ZR�Qp�P�Z���_bTVU�T \kZQadrEp-_bTdi�U#K

OQ}oTV�-\j]`U�iN} �-iNt%�+]"_badcbZ \�� iNsE\VakZ�PNP�U9� �CU �� _bTdc�� _�v iNP adTkZ��bZ�iNPNP�UYq-_b[j[IU�qkroi� [ji�\daVUQZb\V\di�\dakZ�s7a+WEZ�sE\+P�� iNsEWLnE\dadTdi�U#Ki�+s0M�X��#v�#iNPE}oTVp�}:Z�TVU�nosEU [ Z � adTdi�\VU U�sj}orYL\di��7nEU�~P�� nosoiN�9U�TV\ViNaVpQWoUQ��_bTdsEU�PNP��7n`� iNP:_��oadi�U�s4a�nosEUQZ�sosEp-U^}oPNnE\+akZ�TVW�KYe PEadTkZ��bZ�iNPNP�UQU�sE\dnoiNaVUWEZ�sE\�P���ZbWL[jiNsoi�\VadTkZ�adi�_bs q-_b[j[IUJ}or7�\di�q�i�U�s Zb\V\di�\dakZ�s7a�Ko�CUjM�XLM#M!��M�XLM { �oiNP`U-\da�p-WLi� aVU�noT�WLn ��=>.?6@;:G:. �.<= � � 6@2:.?=u-/; ��138%8@4Y6 � 6%;:G�K _ U�sEWEZ�s4a^P�Z _ TVU�[ji���TVU �CnEU�TdTVU� _bsEWLi�Z�P�UF�oiNP`TVU�aV_bnoTdsEUfWEZ�sE\^P�� iNsEWLnE\dadTdi�U�S���T2K_ PNnoa�$ba+�4nEU�WoU�\kU P�Z�sEq-U�T+WEZ�sE\�nosEU q�Z�TdTdi���TVU�Zbq�ZbWop�[ji��4nEUF�#]`_badc#ZC\VU�q-_bsE\kZbq�TkZ�iNa�Y\kU-\JadTkZ-�#Z�nL� WoUjTVU-qVrEU�TVqkrEU }�U�sEWEZ�s4a \VU-\JP�_bi�\diNTV\fU�a��"Wo�-\ M�X��9� ��iNP+}onN�oPNi�Z�p�}oi�\V_� WLi��7nEU�[IU�s4aCWoU-\^Z�Tdadi�q�P�U-\�K@�C� U-\da^Z�iNsE\Vi"�4n`� iNP�TVUzb�noa^P�U~adiNadTVUfWoUJ�~_7q�aVU�noT^�-\^\Vq�i�U�sEq-U-\WoU P�� |QsoiN�9U�TV\diNaVpjWoU ��iNTd[jiNsot#r:Z�[ U�s M�XLM1�LK�]"Z=[ Zd�d_bTdiNaVpjWoU q-U-\fZ�Tdadi�q�P�U-\fZ��bZ�i�U�s7a}�_bnoT \dn��dU�aIP���Z�}o}oPNi�q�Z�adi�_bs WoU-\I[Ip�adrE_�WoU-\=WoUYP�Z }orYL\di��7nEUB� P�ZU�oi�_bP�_bt#i�UF� [���[IU\di�q-U�TdakZ�iNsE\Q\%� U�s p�P�_biNt#s:Z�i�U�s4af�7nEU�P��7nEUf}�U�n3�:q-_b[j[IU q-U�PNnoi8\dnoTQP�ZIq-_bsE\dadTdnEq�adi�_bseWoU\VU-q�adi�_bsE\Qq-_bsoi��4nEU-\ }:Z�T }oPNi�Z�t9U-\^WoU~}:Z�}oi�U�T�K

X9nE\V�4n`� U�s M�X����� ]"_badcbZ sEU )Eaj�4nEU }�_#\VU�T�P�U-\ R _bsEWEZ�adi�_bsE\ WoUYq-U �7noi�\VU�TkZ \kZadrEp-_bTdi�U#K��C� U-\daIt#T � q-U � �QZ�7[I_bsEW _ U�Z�TdP~WoUYP�� nosoiN�9U�TV\diNaVp X9_brosE\ �Q_b}oc�iNsE\ �7n`� iNPq-_bs7adiNs4nEU�TkZf\k_bsjZ��bZ�sEq-p-U#K���U�WoU�Tdsoi�U�Tr��p�sEp#):q�i�Z�iNa Wop ���CW�� nosEU�q-U�TdakZ�iNsEU TVU�sE_b[j[Ip-UF�iNP+p�akZ�iNaf}oTVU�[ji�U�T }oTV_�R U-\k\VU�noTfWoUI�oi�_b[Ip�adTdi�U WoUj\V_bsenosoiN�9U�TV\ViNaVp#KCe P }oT�$bs:Z�iNafP�� noadiNPNi� \kZ�adi�_bs�WoU-\g\dakZ�adi�\dadi��7nEU-\gWEZ�sE\gP�U-\f\Vq�i�U�sEq-U-\K�oi�_bP�_bt#i��4nEU-\ U�af[Ip-WLi�q�Z�P�U-\ U�af�9_bnoP�Z�iNaZ�}o}oPNi��7nEU�T P�U-\ [Ip�adrE_�WoU-\ WoU P�ZQWop�[I_bt#TkZ�}oroi�U(�QP�� p�adnEWoU WoU-\�}�_b}onoP�Z�adi�_bsE\�Z�soiN[ Z�P�U-\S���o}o}0�#v�¤�FXdT2K

Page 25: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� � ��� ������������ ������� � � ��� � � �

_ U�Z�TdP�TVU�[ Z�TV�7n:Z �4n`� iNP�U���i�\VakZ�iNaQnos PNi�U�seU�s4adTVU�\VU-\Cq-U�s4adTVU-\JW�� iNs4aVp�T���aV\gU�aQP���Z�} }oTV_7qkrEU [ Z�adrEp�[ Z�adi��7nEU WoU P�Z �oi�_bP�_bt#i�U }oTV_b}oTVU �^]`_badcbZ��q#� U-\da�Wo_bsEq aV_bnoa�s:Z�adnoTVU�PNP�U# [IU�s4a �4n`� iNP�}oTdiNa q-_bs4akZbq�a Z��9U-q�PNnoi�K�e P4PNnoiL}oTV_b}�_#\kZ^WoU�}�_bno�9_biNT+Zbq-q-p-WoU�T�WLiNTVU-q�aVU�[IU�s7a��\V_bs�P�Z���_bTkZ�aV_biNTVU!�jX9_brosE\ �Q_b}oc�iNsE\��EU�s p-qVr:Z�sot9UgWoUg�4nE_bi"]"_badcbZ��9U�TdTkZ�iNaC\VU-\^Z�T adi�q�P�U-\+}onN�oPNi�p-\�}:Z�T P�U^Wop�}:Z�TdaVU�[IU�s7a�WoUL�oi�_b[Ip�adTdi�U^WoU^P�� nosoiN�9U�TV\diNaVp#Ko�C� U-\da�Z�iNsE\di@�4nEU]`_badcbZjTVUzb�noa P�U~adiNadTVUfWoU _ TV_�R U-\V\VU�noT �C_bsE_bTkZ�iNTVU#K

O�q-U�adaVU }�p�Tdi�_�WoUF�@]"_badcbZ U�noaCP�� i�Wop-U WoU TkZb\V\VU�[!�oP�U�TJ\VU-\^adTkZ��bZ�nL� WEZ�sE\Cnos _bn �7TkZ�t9U � _ U�Z�TdP�P�� U�sEq-_bnoTkZ�t9U�Z�WEZ�sE\Cq-Ug}oTV_��dU�a�Ku�+sEq-_bTVUgR�Z�PNP�Z�iNa �iNP��4nEU ]`_badcbZ adTV_bno�9Unos=TVU��9U�s7nY}�_bnoT�P�Zf}�p�Tdi�_�WoUCWoUQTVp-WEZbq�adi�_bs`KNe P@}oTdiNa Wo_bsEq~q-_bs4akZbq�a^Z-�9U-qCnos�iNsE\dadiNadnoaWoUgTVU-qkrEU�TVqVrEU�[Ip-WLi�q�Z�P�U#K`��U�}�U�sEWEZ�s7a���[���[IU \%� iNP8adTkZ��bZ�iNPNP�Z�iNafZ�P�_bTV\~\dnoT^nosEU p�adnEWoU[ Z�adrEp�[ Z�adi��4nEUeWoU P�Z�[ Z�P�Z�Tdi�Z��\VU-\ adTkZ-�#Z�nL� s`��Z-�#Z�i�U�s4a��7n`� nos iNs4aVp�T���aY[Ip-WLi�q�Z�PPNiN[jiNaVp#K_ U�Z�TdP Z�noTkZ�iNaf}oTVp�R p�TVpj�4nEUI]"_badcbZ�sEUj\%� p�P�_biNt#sEUj}:Zb\fWoUIX9_brosE\ �Q_b}oc�iNsE\���iNP�)EaWo_bsEqC\k_bs=}�_#\V\di��oP�UQ}�_bnoT PNnoi�R�Z�iNTVUQnosEU~_�"�TVU^iNs7aVp�TVU-\V\kZ�s7aVU#KEO^P�_bTV\ �7nEU~]`_badcbZ�Z��bZ�iNa�4nEU�P��7nEUJ}�U�n�Wop�P�Z�i�\V\Vpf\V_bs0}:Z�}oi�U�TQ\dnoT�P�Z�[ Z�P�Z�Tdi�Z�@iNP"Z�}o}oTdiNaQ�7nEUJP�UfWop�}:Z�TdaVU�[IU�s7aWoU _ U�Z�TdPCqVrEU�TVqkr:Z�iNa=nos }oTV_�R U-\V\VU�noT�K��JZ�sE\jnos }oTVU�[ji�U�T aVU�[j}E\��+iNP sEU�R noaIt#nE��TVUU�s4adrE_bnE\di�Zb\daVU~q�Z�T q-U�P�Z \diNt#soi�)@Z�iNa P���Z��:ZbWo_bs�W�� nos�[IU�iNPNP�U�noT�\kZ�P�Z�iNTVU#K _ U�Z�TdP�sE_bs }oPNnE\sEU��9_�4Z�iNa }:Zb\ b�ZeW�� nos adTV�-\c��_bsB_�U�iNP �+q�Z�T q-UY\VU�TkZ�iNa Z�PNP�pB� P�� U�sEq-_bs4adTVU WoUY\V_bs}oTV_�R U-\V\k_bTkZ�a rE_bsE_bTkZ�iNTVUF��adiNadTVU�}oPNnE\jp�P�_bt#i�U�nL��K �Cp�Z�so[I_biNsE\���}�U�n WoU aVU�[j}E\IZ�}oTV�-\_ U�Z�TdPfU�sEq-_bnoTkZ�t9U ]"_badcbZ ��}�_#\dadnoP�U�T�U�a�P���Zb\V\dnoTVU _�P q�i�U�nE\VU�[IU�s4a �4n`� iNPfZ�noTkZ�iJP�Z}oP�Zbq-U�S���L}o}O������LM#T2K

��i�U�s �7nEU�P�ZeR Z�b-_bs Wo_bs4a�]`_badc#Z�Z Z�s:Z�P��\kp�P�U [I_bsEWoU0s:Z�adnoTVU�P~U-\da nosoi��4nEUF�U�PNP�U0q-_bTdTVU-\d}�_bsEW Z�n }�_biNs7a WoU0�7nEU WoU9��U�Z�nEq-_bno} WoU0\kq�i�U�s4adi�):�7nEU-\ WEZ�sE\ WLiN�9U�TV\Wo_b[ Z�iNsEU-\!� P�� p�}�_7�7nEU#KCO�U�P�_bs�U�nL�:��sE_bs \VU�noP�U�[IU�s4a P�U�aV_bnoagWop�}�U�sEW W�� nosEUI\kZ�iNsEUiNs4aVU�TkZbq�adi�_bseWoU-\^}:Z�Tdadi�U-\��E[ Z�i�\QiNP8U��oU�TVq-Ugnos q-_bs4adT�$bP�U�\dnoTQU�PNP�U-\%K �Qs0TVU�adTV_bno�9U q-U-\sE_badi�_bsE\�U�s=}oroiNP�_#\V_b}oroi�UF�oq-_bsEq-U�Tds:Z�s7a P�Zg}�U�TVq-U�}oadi�_bs�\VU�sE\k_bTdi�U�PNP�UF�7P�Z q-_bsE\Vq�i�U�sEq-U~U�aP���Z�}o}oTVU�s4adi�\V\kZ�t9U �@q-Uf\V_bs7a�PM��adTV_bi�\ U��oU�[j}oP�U-\^WoU-\�adrE��[IU-\^adTkZ�iNaVp-\ }:Z�T^]`_badc#Z�S � T2K

O^n=�7n WLn=Wop��9U�P�_b}o}�U�[IU�s4a WLn=sE_b[K�oTVU~WoU }onN�oPNi�q�Z�adi�_bsE\ Z�4Z�s4a adTkZ�iNa��fP�Z[�oi�_� }orYL\di��4nEUF��]`_badc#Z \�� iNsE�7noi�p�akZLK0e P q�TkZ�iNt#s:Z�iNaI�4nEU��7nEU�P��4n`� nos s`��Z�s4adi�q�iN}�UY\VU-\ i�Wop-U-\�Ke2P( U�noaYU�s4adTVU�Z�noadTVU-\�nosEUep�adnEWoUe\dnoT�P���Z�}o}oPNi�q�Z�adi�_bs WoU-\�[ Z�adrEp�[ Z�adi��4nEU-\ Z�nL�}oTV_��oP���[IU-\JWoU q�TV_bi�\V\ Z�sEq-U U�a~WoUg[I_bTd}orE_bP�_bt#i�UF��nos Z�}o}�U�P�� nosEU }oPNnE\Qt#TkZ�sEWoU�q-_� _b}�p�TkZ�adi�_bs�U�s4adTVU�}orYL\di�q�i�U�sE\�U�aC�oi�_bP�_bt#i�\daVU-\�U�a�nosgPNiN��TVU�WoU\�oi�_b}orYL\di��7nEU s`� iNsE\di�\VakZ�s4a�4nEUQ\dnoT+P�ZJ}or7�\Vi�_bP�_bt#i�U#K 8_bnoa�q-U�P�Zf}�_bnE\k\kZ])Es:Z�P�U�[IU�s7a ]"_badcbZ!�gZbq-q-U�}oaVU�T�P�Zf}oP�Zbq-UWoUg}oTV_�R U-\V\VU�noT�K>e P"}:Z�TdadiNaJWo_bsEqc� ��Z�PNadiN[I_bTVU }�_bnoTQP�Z=TVp-WEZbq�adi�_bseWoU-\K* 476 � 6%;:G:,F�q-UJ�4noi�PNnoi`}oTVU�sEWLTkZIWoU�nL��Z�sE\KS���o}`K����dT2K

]`_badc#Z }�U�sE\kZ�iNaJ�7nEUF��}�_bnoTJ�7nEU�P�U-\K* 476 � 6%;:G:,�TVU�sEq-_bs7adTVU�s4agnose��TkZ�i \dnEq-q-�-\��iNP"R�Z�PNP�Z�iNaC�7nEU P�U-\Q\Vq�i�U�s7adi�):�4nEU-\J\k_bi�U�s4aJq-_bs4�#Z�iNsEq�nE\JWoUgP�� noadiNPNiNaVp�W�� nos aVU�P�_bno��TkZ�t9U#K

Page 26: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

�#m ��������� ������� � � � ��� � ����� � � ���������

e P�)Ea P�� U�TdTVU�noTIWoU�\VU aV_bnoTdsEU�Tj}oTVU-\V�7nEU�U��Lq�PNnE\ViN�9U�[IU�s4aI�9U�TV\�P�U [I_bsEWoU�}orYL\di��7nEUZ�n Wop�adTdiN[IU�s4aYWoU�P�Z5�oi�_bP�_bt#i�U#K�OQP�_bTV\ �7n`� O�WEZ�[I\ PNnoiQ\dnot#t9p�TkZ�WoU-\I}onN�oPNi�q�Z�adi�_bsE\WEZ�sE\�P�Z TVU���nEUJ\Vq�i�U�s4adi�):�7nEUK* =>E04YE���\�E]`_badc#Z \�� iNsE�4noi�p�akZ�iNaQWLn=R�Z�iNa��4nEU~\V_bsYPNiN��TVUs`� iNs7aVp�TVU-\V\kZ�iNaQU-\k\VU�s4adi�U�PNP�U�[IU�s7aC�7nEUJWoU-\ p-q-_bP�_bt#i�\daVU-\�K

O�P�� U��oq-U�}oadi�_bs W�� nosg\dakZ�adi�\Vadi�q�i�U�sgWoU �CZ�T�� Z�TVWg�7noi9WEZ�sE\�P�U }�p�Tdi�_7WLi��7nEU ��=>.?6@;3=>6q-_b[j}:Z�TkZ P�U-\W* 4Y6 � 6%;:G:,B� � iNsosoi�U�P�� �QnoTV\V_bs3��P�U PNiN��TVU U�noajWoU���_bsE\�q-_b[j[IU�s akZ�iNTVU-\�KN� Z�PNt#TVpJq-U�P�Z�o]`_badc#Z s`� p�akZ�iNa�}:Zb\ q-_bs4aVU�s7a��oiNP"Z-�#Z�iNa�P�U~\VU�s4adiN[IU�s7aQ�7n`��Z�nEq�nosWoU \VU-\QP�U-q�aVU�noTV\~s`��Z-�#Z�iNa~\ Z�i�\di"P�� U-\V\kU�sEq-U WLn PNiN�7TVU#K�]`U TVU-\d}�_bsE\kZ��oP�U�WoUgP�Z }onN�oPNi�q�Zd adi�_bs PNnoi Zb\V\dnoTkZ �7nEU P�U-\��9U�s4aVU-\Ip�akZ�i�U�s7aITkZ�i�\V_bsos:Z��oP�U-\�K O }:Z�TdadiNTIWoUeM�X����� U�PNP�U-\R�Z�i��oPNiNTVU�s4a�� nE\V�7n`� �gq-UC�4nEU~q-U�adaVUC}oTVU�[ji���TVU~p-WLiNadi�_bs�\V_biNa�}oTVU-\V�4nEU~p�}onoi�\Vp-UCU�seM�X { �LK

|QsEUjR _bi�\fP�ZYTVp-WEZbq�adi�_bs WoU-\c* 476 � 6%;:G:,BE�� � ���,�.<=@-04 ].<E/47E��� ZbqkrEU��9p-UF�]`_badc#Z~Wop�P�Z�i�\V\kZCq-U Wo_b[ Z�iNsEU Z�n�}oTV_�)Ea+WoU�P�ZQWop�[I_bt#TkZ�}oroi�UF�i�oTkZ�sEqVrEU WEZ�sE\�P�Zb�7nEU�PNP�UiNP�\VUI)EagnosEU=U��Lq-U�PNP�U�s7aVU TVp�}onoakZ�adi�_bs`K ��U�TdakZ�iNsE\ }:Z�TdP�U�s4a�WoUjPNnoi�q-_b[j[IU WLn �CU � aV_bs�WoUIP�ZYWop�[I_bt#TkZ�}oroi�UF�8U�a��8Z�n�� _bnoTVW�� r7noi�U�sEq-_bTVUF�8\VU-\JadrEp-_bTdi�U-\g\k_bs4agP�Z+�:Zb\VUIWoUsE_bno�9U�PNP�U-\^iNs4�9U-\dadiNt4Z�adi�_bsE\�K

]"U-\K* 4Y6�� 6@;:G:,�_bs4aJp�aVp iN[j}�_bTdakZ�s7aV\K�=WoU�nL� adiNadTVU-\�K���� nosEU }:Z�Tda��@iNP�\JZ�}o}�_bT aV��TVU�s7aJnosEUK�:Zb\VU�q-_bsEq�TV��aVUc� P�� p-q-_bP�_bt#i�U�s:Z�i�\V\kZ�s4aVUF��W���Z�noadTVU }:Z�Tda���P�U [I_7Wo��P�U [ Zd adrEp�[ Z�adi��7nEU }oTV_bi�U# �}oTVp-WEZ�aVU�noTg�4nEU ]`_badc#ZI}oTVp-\kU�s4aVUIZ�adadiNTkZ nosEU t#TkZ�sEWoUjZ�adaVU�s7adi�_bsQ Z�n��d_bnoTVW�� r4noi^U�sEq-_bTVU�iNP U-\Va�q-p�P��&�oTVU0U�ajaV_bn�� _bnoTV\jp�adnEWLi�p#K���U R noa�nosEU�t#TkZ�sEWoUq-_�� sEq�i�WoU�sEq-UF��Z�n�[���[IUJ[I_b[IU�s4a��@WoU-\�TVp-\dnoPNakZ�aV\Q\diN[jiNP�Z�iNTVU-\�R noTVU�s4a }onN�oPNi�p-\!S���o}o}{ �� { XdT2K����� ��� #��� � � �#���

hCiNaV_=h+_bPNaVU�TdTkZ�:[ Z�adrEp�[ Z�adi�q�i�U�seiNakZ�PNi�U�seZ�P�_bTV\R�jP�Z a���aVU WoUgP�Z qkr:Z�iNTVU WoUg}orY7 \di��7nEU�[ Z�adrEp�[ Z�adi��4nEUL� �^_b[IU�U-\dar� P�� _bTdiNt#iNsEU WoU�q-U�adaVU�}onN�oPNi�q�Z�adi�_bs`K#]"U �onoa�WoU�\VU-\TVU-qkrEU�TVqVrEU-\ p�akZ�iNa�WoU�adTkZ�sE\dR p�TVU�T�P�U-\ q-_bsEq-U�}oaV\�WoU�P�ZJ[Ip-q�Z�soi��7nEU^q�P�Zb\V\Vi��4nEU �JP�Z]�oi�_� P�_bt#i�U#Kie PEq-_b[j[IU�s%b�ZJ}:Z�T P�U-\+p-�4n:Z�adi�_bsE\ WLi�"�p�TVU�s7adi�U�PNP�U-\�_bTVWLiNs:Z�iNTVU-\�U�a P�U-\+p-�7n:Z�adi�_bsE\iNs7aVp�t#TV_� 2WLi�"�p�TVU�s7adi�U�PNP�U-\����7n`� iNP�noadiNPNi�\kZeZd)Es W�� p�P�Z���_bTVU�TjnosEU�[Ip-q�Z�soi��4nEU0WoU-\IZb\V\V_� q�i�Z�adi�_bsE\ �oi�_bP�_bt#i��4nEU-\�� }onoi�\ iNPQU�noa TVU-q-_bnoTV\�Z�nL� p-�4n:Z�adi�_bsE\ R _bsEq�adi�_bsosEU�PNP�U-\=}�_bnoTZdPIsEU�Tf\VU-\fZ�s:Z�P�L\VU-\�K3O�_bse}�_biNs4aJWoU���nEU p�akZ�iNafWop�aVU�Td[jiNsoi�\VaVUF�`q-U��4noi q-_bTdTVU-\d}�_bsEWaV_bakZ�P�U�[IU�s7a]� P�� nE\kZ�t9UgWoU-\ _bnoadiNP�\^�7noi`_bs4a^p�aVpfP�U-\ \di�U�sE\

OQP�_bTV\��4nEU W���Z�noadTVU-\�}oT�$bs:Z�i�U�s4ajnosEU=��i�\di�_bs�}oTV_��:Z��oiNPNi�\VaVUF� _bn WLn [I_biNsE\j\dakZd adi�\dadi��7nEUF� h+_bPNaVU�TdTkZBZbq-_�_bTVWEZ�iNa�[I_biNsE\�WoUeq�TVp-WLiNa � [Ip�adrE_7WoU �7n`� � nosEU�Z�s:Z�P�L\VUiNs)EsoiNaVp-\diN[ Z�P�U#K e P:U-\da W���Z�iNPNP�U�noTV\�iNs4aVp�TVU-\V\ Z�s4a WoU TVU�[ Z�TV�7nEU�T��7nEUF�9P�_bTV\V�7n`� iNPEs`��Zf�7nEUadTVU�i ��U Z�sE\ U�a^�7n`� iNP"U-\V\kZ�9UfWoUJWop�aVU�Td[jiNsEU�TQP�Z adTkZd�dU-q�aV_biNTVUgW�� nos0}oTV_�� U-q�adiNP�U WEZ�sE\ P�Uqkr:Z�[j} t#TkZ��7iNakZ�adi�_bsosEU�P`q-_b[!�oiNsEpJWoU^P�Z "U�TdTVU~U�a WoU^P�Z ]�nosEUF��P�Zg\k_bPNnoadi�_bs �7n`� iNP�U�� roi���UJ\VU(�:Zb\VU~\dnoT nosY}:Z�TdadiNadi�_bsosEU�[IU�s4a~WLn�aVU�[j}E\ U�sY}�U�adiNaV\ iNs7aVU�Td�bZ�PNP�U-\��o}�U�sEWEZ�s7aP�U-\V�7nEU�P�\�P�Z R _bTVq-UJU-\da q-_bsE\di�Wop�TVp-U q-_bsE\dakZ�s4aVUgU�a P�Z adTkZd� U-q�aV_biNTVUg}:Z�TkZ���_bPNi��7nEU�S��dT2K

h+_bPNaVU�TdTkZCq-_b[j[IU�s%b�Z]��adTkZ��bZ�iNPNP�U�T�\dnoT�P�U-\8W�s:Z�[ji��7nEU-\8WoU�}�_b}onoP�Z�adi�_bsE\�P�_bTV\V�7nEU|Q[K��U�TdaV_ ��� OQsEq-_bs:Z�:\V_bs9��U�Z�n �)EP�\��:PNnoi`}oTVp-\VU�s4akZInos0}oTV_��oP���[IU Zz4Z�s4aCadTkZ�iNaCZ�nL�

Page 27: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� � � � ��� ��� � � � ��������� �9�

\dakZ�adi�\dadi��7nEU-\ }�_bi�\V\V_bsoi���TVU-\^WoUgP�� O�WLTdi�Z�adi��4nEU#K�e P`}onN�oPNi�ZI\VU-\^}oTVU�[ji�U�TV\^TVp-\VnoPNakZ�aV\^\dnoTP�U-\ �EnEq�adn:Z�adi�_bsE\^WLn�sE_b[!�oTVUgW�� iNsEWLiN��i�WLnE\^W�� U-\d}��-q-U-\ U�s0q-_br:Z��oiNakZ�adi�_bs U�s�M�X��#mLKe2P�aVU�s7akZ�oU�s�R Z�iNa��LW���Z�}o}�_bTdaVU�T nosEU~U���}oPNi�q�Z�adi�_bs0\kq�i�U�s4adi�):�7nEUgZ�nL� _��E\VU�Td�#Z�adi�_bsE\WoU ��� OQsEq-_bs:Z Z@P�U-\C}�_bi�\V\V_bsE\~WoU aa�}�U }oTVp-WEZ�aVU�noTg_bs7aC��n P�U�noT~sE_b[K�oTVUIZ�not#[IU�s aVU�T^\VU�sE\di��oP�U�[IU�s7aQU�s0O�WLTdi�Z�adi��7nEUf}�U�sEWEZ�s4aQP�Z _ TVU�[ji���TVU��CnEU�TdTVU[� _bsEWLi�Z�P�UF�@Z�P�_bTV\�4nEUQP�U^sE_b[!�oTVUJWoU^P�U�noT }oTV_bi�U-\��7PNnoi �LWLiN[jiNs4n:Z�iNa�Ko�~UC}oPNnE\���U�a�q-U�P�Z \dnE\Vq�iNakZ�iNa�P�� p�aV_bs sEU�[IU�s4a��8P�_bTV\V�4nEU=P�ZY}>�-qVrEU=TVU�}oTdiNa����YP�Z�)Es WoUIP�Z�t#nEU�TdTVUF�"iNP�s`� �Z-�#Z�iNa }:Zb\ }oPNnE\WoUJ}�_bi�\k\V_bsE\��o\diN[j}oP�U�[IU�s7aQP�U-\^}oTV_b}�_bTdadi�_bsE\QU�s4adTVUfP�U-\QWLi�"�p�TVU�s7aVU-\QU-\V}��-q-U-\CZ-�#Z�i�U�s4ap��9_bPNnEp-U-\KS { �L}`K � T2K

_ _bnoT ��� O^sEq-_bs:Z�4P���Z�TdT���a�WoU^P�ZJ}>�-qVrEUF�4[���[IU^aVU�[j}�_bTkZ�iNTVUF�7\%��Zbq-q-_b[j}:Z�t#sEU^iN[I [ Z�sE�4n:Z��oP�U�[IU�s7a W�� nos�Wop�}oP�Zbq-U�[IU�s7a^WoUCP�� p-�4noiNPNi��oTVU(�oi�_bP�_bt#i��7nEUJU�s R�Z-�9U�noT�WoU-\ U-\ }��-q-U-\ }oTVp-WEZ�adTdi�q-U-\�KL]`U }oTVU�[ji�U�T [I_�Wo��P�U^�7noiER noa�p�akZ��oPNiNa Wop-q�TdiNa�iNs7aVU�TkZbq�adi�_bsE\�U�s4adTVUWoU�nL� U-\d}��-q-U-\cZ�}oTV_bi�UIU�aJ}oTVp-WEZ�aVU�noT���Z�}oTV�-\f�4nE_bi iNP�R noaJt9p�sEp�TkZ�PNi�\VpW� nos�sE_b[K�oTVU�4nEU�P�q-_bsE�7nEUfW�� U-\d}��-q-U-\�Q P�Z�}:Z�TdiNaVpfWoUgq-U~sE_b[K�oTVU �d_bn:Z�s4a^nos�T�$bP�UfWop�aVU�Td[jiNs:Z�s4a�K

h+_bPNaVU�TdTkZ�U�aJ� � OQsEq-_bs:Z=}onN�oPNi���TVU�s4a �\6��>E/;3,D,�132 4F-mG ���%E02:.<6 � -G ��� �L- �G>.��0136 A/6 4i- 471:G>G:6 8%E0132 4F- /.<6�U�s M�X��LM _�v iNP�\C}oTVp-\VU�s4aVU�s7a~P�U�noTV\Q[I_�Wo��P�U-\U�agq-_bsEq�PNnE\di�_bsE\�K8]`UI\dn��dU�agWLn PNiN��TVUc U-\daf}oTVp-\VU�s7aVp q-_b[j[IUIP�� p�P�p�[IU�s7a q-U�s7adTkZ�P�U�aiNsEWLi�\d}�U�sE\ Z��oP�UfWoU~aV_bnoaVUfp�adnEWoUf\dnoT�P�U-\CZb\V\V_�q�i�Z�adi�_bsE\ �oi�_bP�_bt#i��7nEU-\�K

�JZ�sE\�\VU-\ � 6��>E0;3,F� h+_bPNaVU�TdTkZ p�sE_bsEq-UeadTV_bi�\YP�_bi�\�� �9_biNT { K��LK��LK^]"Z adTV_bi�\di���[IUq-_bTdTVU-\d}�_bsEW�aV_bnoa � R Z�iNa � P�Zj�7i�\di�_bs�WoUg� � OQsEq-_bs:Z+Z

O7i:P�� _bs�Wop�adTdnoiNa�P�U-\�WoU�nL� U-\d}��-q-U-\+nosoiSR _bTd[Ip�[IU�s4a U�a�}oTV_b}�_bTdadi�_bsosEU�PNP�U# [IU�s4a�Z�nL�IsE_b[K�oTVU-\ WoU^P�U�noTV\ iNsEWLiN��i�WLnE\��7P�ZJ[I_�9U�sosEUCWLn sE_b[K�oTVUCWoU-\iNsEWLiN��i�WLnE\QWoU P�� U-\d}��-q-UgWop��9_bTVp-U q�TV_ � aCU�a~q-U�PNP�UgWoU P�� U-\d}��-q-UgWop��9_bTkZ�s4aVUWLiN[jiNs4nEU S � �o}`K��#mdT2K

��U aa�}�UQWoUQq-_bsEq-_bTVWEZ�sEq-ULQ�P�_bTV\V�7nEUF�9}:Z�TdakZ�s7a W�� r77}�_badrE�-\VU-\���TkZ�i�\VU�[K�oP�Z��oP�U-\��7P�U-\q-_bsEq�PNnE\di�_bsE\~adrEp-_bTdi��4nEU-\g\V_bs4afq-_bs)ETd[Ip-U-\J}:Z�T~P�U-\J_��E\VU�Td�#Z�adi�_bsE\���U��L}�p�TdiN[IU�s7akZ�P�U-\Q p�akZ�iNa�}�_bnoTYh+_bPNaVU�TdTkZ P�Z q�P�U�R�WLn Wop��9U�P�_b}o}�U�[IU�s4a \Vq�i�U�s4adi�):�7nEU#K _ _bnoT0PNnoi � iNP\���Z�t#i�\V\kZ�iNagU�s R Z�iNagWoUIP���Z�}o}oPNi�q�Z�adi�_bs q-_bsEq�TV��aVU ��P�Z��oi�_bP�_bt#i�U WoU-\J[Ip�adrE_7WoU-\ WoUjP�Z}orYL\di��4nEU~[ Z�adrEp�[ Z�adi��7nEUgWLnB'(e 'CU~\di��-q�P�U#K

�^U�[ Z�TV�7nE_bsE\8�7nEU P�U adTkZ-�#Z�iNP�WoU h+_bPNaVU�TdTkZ }�U�noa��%\V_bnE\8q-U�a�Zb\d}�U-q�a �PM��%\%� iNsE\Vq�TdiNTVU �q-_bs4adTVU# 2q-_bnoTkZ�s4afWoUgP�ZjaVU�sEWEZ�sEq-U t9p�sEp�TkZ�P�UI�jP�� p�}�_7�7nEUWZ:P�U-\ [ Z�adrEp�[ Z�adi��7nEU-\J\V_bs4aU�s U#"�U�aj}:Zb\V\Vp-U-\jWLn \VakZ�adnoa�WoU�P�Z�sot#n:Z�t9U0WoU=P�Z s:Z�adnoTVU9� q-U�PNnoi W�� U�sE\kU�[K�oP�U�WoUq-_bsos:Z�i�\V\kZ�sEq-U-\ ���:Zb\VU~W���Z �Li�_b[IU-\ q-_b[j}�_#\VpJWoU~adrEp-_bTdi�U-\ R _bTd[gnoP�p-U-\ iNsEWop�}�U�sEWoU�[I [IU�s4a WoU-\ Z�noadTVU-\�\Vq�i�U�sEq-U-\�K9�~�-\ P�_bTV\��bP�� iNs4aVp�T���a aV_bnoa�WoU�[���[IU�adTV�-\ iN[j}�_bTdakZ�s4a��4nEU}�_bTdakZ�iNa P���e akZ�PNi�U�s�}�_bnoT�P�U-\ q-_bs)ETd[ Z�adi�_bsE\CU���}�p�TdiN[IU�s4akZ�P�U-\Q}�U�noa�}:Z�TkZ � adTVU Wop-\dnEU�a�K

h+_bPNaVU�TdTkZ��Z�}o}oPNi��7n:Z�s4a P�U-\f[Ip�adrE_�WoU-\ WLn�']ea'~Uj\di�p-q�P�UF�"\kqVrEp�[ Z�adi�\kZ�iNa P�Z 2[ Zd adrEp�[ Z�adi�\kZ�adi�_bs �fW�� nosEUf\Vq�i�U�sEq-UJU�s��7n:Z�adTVUJ}or:Zb\VU-\��oWo_bs7a P�U-\�adTV_bi�\�}oTVU�[ji���TVU-\Q\V_bs4aTkZ�adi�_bsEU�PNP�U-\�U�a�P�Z �7n:Z�adTdi���[IUF�oZ�}o}oPNi��4nEp-U#Ko|QsEU R _bi�\�P�U-\��4n:Z�adTVU-\�}or:Zb\VU-\�U#"�U-q�adnEp-U-\��iNP`R�Z�PNP�Z�iNa^TVU��9U�soiNTR�jP�Zj}oTVU�[ji���TVU�U�aQTVU-q-_b[j[IU�sEq-U�T�K�� Z�i�\Q}�_bnoT~Z�}o}oPNi��4nEU�T~q-Uf}oTV_� q-U-\V\dnE\c� P�ZD�oi�_bP�_bt#i�UF� h+_bPNaVU�TdTkZ sEU }�_bno�#Z�iNaI\VU }:Zb\V\VU�TjWoU��oi�_bP�_bt#i�\daVU-\���P�U-\jWoU�nL�

Page 28: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

�#v ��������� ������� � � � ��� � ����� � � ���������

}oTVU�[ji���TVU-\~}or:Zb\VU-\~q-_bsE\di�\VakZ�s4a[� p�adnEWLi�U�T~P�U-\Q}oTV_b}oTdi�p�aVp-\JWLn \��\daV��[IU U�aR�IP�U-\CWop# q�TdiNTVUfU�s0aVU�Td[IU-\�[ Z�adrEp�[ Z�adi��4nEU-\%K

8_bnoaVU�R _bi�\���iNPQsEU�}�_bno�bZ�iNa�q-_bPNP�Z���_bTVU�T�Z��9U-q0P�U }oTVU�[ji�U�T �oi�_bP�_bt#i�\VaVU �9U�s7n �7nois`��Z�noTkZ�iNajTdi�U�s q-_b[j}oTdi�\�� \VU-\ adrEp-_bTdi�U-\�[ Z�adrEp�[ Z�adi��4nEU-\%K �C� U-\da�� q-U [I_b[IU�s7a �PM��7n`� iNs4aVU�Td��i�U�s4a hCP�ZbWLiN[jiNT �Q_#\dadi �-iNs H M�v#v��� M�X#m�� J � nos 2[ Z�adrEp�[ Z�adi�q�i�U�s�s:Z�adnoTkZd PNi�\daVU � � q-_b[j[IU iNPJ\VU Wop-q�TdiNa PNnoi� �[���[IU#K���� nosEU }:Z�Tda���iNPJZ�p�adnEWLi�p P�U-\�p-�7n:Z�adi�_bsE\iNs7aVU�t#TV_� 2WLi�"�p�TVU�s7adi�U�PNP�U-\��`q-U �4noi�sEU \kU�TkZj}:Zb\CsEp�t#PNiNt9U�Z��oP�U }�_bnoTCP�Z q-_b[j}oTVp�rEU�sE\Vi�_bsWoU-\ adTkZ��bZ�nL�YWoUCh+_bPNaVU�TdTkZ�oU�a��oW���Z�noadTVU~}:Z�Tda��oq-_b[j[IU~h+_bPNaVU�TdTkZ�LiNP�}�_bTdaVU~nos�t#TkZ�sEWiNs7aVp�T���aQZ�n�}oTV_��oP���[IUfWoU~P�� p��9_bPNnoadi�_bs`K%e PY�d_bnEU�TkZ nosYT�$bP�UCiN[j}�_bTdakZ�s7a^WEZ�sE\�P�U~Wop��9U# P�_b}o}�U�[IU�s4a WoU-\+TVU-qVrEU�TVqVrEU-\�WoU h+_bPNaVU�TdTkZLK9�~�-\ \kZQ}oTVU�[ji���TVU�}onN�oPNi�q�Z�adi�_bsE\��4h+_bPNaVU�TdTkZ\VUJ\k_bnEq�i�Z�WoU~P�ZK��_bsosEUfWLi�"�nE\di�_bs�WoUJ\VU-\ TVp-\dnoPNakZ�aV\ }:Z�Td[ji�P�U-\ �oi�_bP�_bt#i�\daVU-\�K:��U�}�U�s WEZ�s7a���P�U-\ TVU�P�Z�adi�_bsE\jU�s7adTVU�P�U [ Z�adrEp�[ Z�adi�q�i�U�s U�a P�U=[jiNPNi�U�n �oi�_bP�_bt#i�\daVU sEU=R&noTVU�s7a}:Zb\ U��oq-U�PNP�U�s4aVU-\�S { �L}o} � ��T2K��� � ����� � ������� �������

�+sBM�X��#m��]`_badc#Z�p-q�TdiNa!�=h+_bPNaVU�TdTkZ�Z�n \dn��dU�afW�� nos�Z�Tdadi�q�P�Uj}onN�oPNi�p�}:Z�Tfq-UjWoU�T soi�U�T��7}:Z�TVq-UQ�7nEU P�U-\�\dn��dU�aV\ adTkZ�iNaVp-\�\VU^TVU�adTV_bno�9U�s4a U�s }:Z�Tdadi�UCWEZ�sE\ \VU-\�* 4Y6 � 6%;3G>,i�[���[IUj\di�P�Z=R�Z�b-_bs Wo_bs4aJiNP�\J\V_bs7agZ���_bTVWop-\gWLi�"���TVUj�7nEU�P��4nEU�}�U�n`K�h+_bPNaVU�TdTkZ�PNnoi�TVp# }�_bsEW U�sIPNnoi�ZbWLTVU-\V\kZ�s7a�\V_bs [Ip�[I_biNTVUQq-_b[j}oP�U�a�Wo_bs4a�Z-�#Z�iNa�p�aVp^U��LadTkZ�iNa P���Z�Tdadi�q�P�UCU�s�7nEU-\dadi�_bs3�oU�a q#� U-\da�h+_bPNaVU�TdTkZj�7noi �LP�U~}oTVU�[ji�U�T���R�Z�iNa�TVU�[ Z�TV�4nEU�T^q�P�Z�iNTVU�[IU�s7aQ�7nEU~\VU-\q-_bsEq�PNnE\di�_bsE\ q-_�� sEq�i�WoU�s7a Z-�9U-q q-U�PNP�U-\ WoU-\ * 476 � 6%;:G:,#K��~�-\8P�_bTV\�� qkr:Zbq�nosj�bZQp�adnEWLi�U�TP�U-\ adTkZ��bZ�nL�IWoU�\V_bsjsE_bno�9U�Z�n q-_bPNP�Z���_bTkZ�aVU�noT�U�a aVU�s4aVU�T WoU P�U-\�R Z�iNTVU�q-_bsos:Z � adTVUQWEZ�sE\\V_bs�U�s7aV_bnoTkZ�t9U \kq�i�U�s4adi�):�7nEU#K

h+_bPNaVU�TdTkZ��q-_b[j[IU=iNP P�� p-q�TdiNa�WEZ�sE\gnosEU P�U�adadTVU �0]`_badcbZ0WEZ�aVp-U WLn �7iNsot#a [ Z�TV\M�X��#m�Eq-_bsE\Vi�Wo��TVUg�7nEUJWoU-\^q-_bsEq�PNnE\di�_bsE\Q\diN[jiNP�Z�iNTVU-\^_��oaVU�s7nEU-\QWoUJadTkZ��bZ�nL�YiNsEWop�}�U�s WEZ�s7aV\�U�a�}:Z�TdakZ�s7a WoUQ}�_biNs4aV\ WoU^��nEUCWLi�"�p�TVU�s7aV\�U�s q-_bs)ETd[IU�s7a�P�Z � nE\daVU-\V\VUF�7q-UQ�7nEUq-_bs)ETd[IU�TkZ ]`_badc#Z WEZ�sE\ nosEU WoU q-U-\gP�U�adadTVU-\�K�]`_badc#Z�R Z�iNa TVU�[ Z�TV�4nEU�TjWEZ�sE\ P�� nosEUWoUJ\VU-\ P�U�adadTVU-\ �4n`��Z�nY��n�WoU~P�� iN[j[IU�sE\VUfq-_b[j}oP�U��LiNaVpfWoU-\ \��\daV��[IU-\ }or7�\Vi��4nEU-\ �7nEUP�Z �~Z�adnoTVUf}oTVp-\kU�s4aVUF�EiNP�U-\da�sEp-q-U-\V\kZ�iNTVUg}�_bnoT^}�_bno�9_biNTQ}oTV_bt#TVU-\V\VU�T��:WoUgq-_b[j[IU�sEq-U�T}:Z�T�WoU-\ U��LU�[j}oP�U-\ [I_�WoU-\daVU-\ U�a�t#TkZ�sEWoU�[IU�s4a^\diN[j}oPNi�):p-\%K��Qs TVU�[ Z�TV�7nEU~Z�iNsE\Vi��7nEU� nE\V�7nEUJWEZ�sE\^WoU-\^p-qkr:Z�sot9U-\Qp�}oi�\VaV_bP�Z�iNTVU-\QU�s7adTVUft#TkZ�sEWo\^\kq�i�U�s4adi�):�7nEU-\��:WoU-\^sE_badi�_bsE\�7noi�}�U�no�9U�s7a^sE_bnE\ \kU�[K�oP�U�TQp���i�WoU�s4aVU-\Q_bs4a PNi�U�n W�����adTVUJ}oTVp-q�i�\Vp-U-\�K

�C_baV_bsE\"�7nEU q-U�adaVU q-_b[j[ nosoi�q�Z�adi�_bs U�s7adTVU+P�U-\8WoU�nL�~\Vq�i�U�s7adi�):�4nEU-\�Z sE_bakZ�[j[IU�s7aU�n=}�_bnoT�q-_bsE\Vp-�4nEU�sEq-UQWoU TkZ��7iN�9U�T�P�U�noT iNs7aVp�T���a�}�_bnoT P�U�noT�\dn�� U�a�W�� p�adnEWoUQq-_b[j[ nos`K�Qs TVU-\k\VU�s4afq-U�P�Z�Z�n Wop&�onoa U�aK� P�Z )Es WoU�}oTVU-\V�7nEUjqVr:Zb�7nEUIP�U�adadTVUF��}:Z�TfU��oU�[j}oP�UP�_bTV\V�7nEUf]`_badc#Z�p-q�TdiNaKZ

� _bnoT P�U�adaVU�T _�R �C_��9U�[!��U�T2����adr r:Zb\�t#iN�9U�s [IU�t#TVU�Z�a�}oP�U�Zb\dnoTVU#K e2a+r:Zb\�9U�Tp WoU#)EsoiNadiN�9U�P� U�sEq-_bnoTkZ�t9U-W�[IUjaV_YTVU�sEU � adrEU ��_bTdc ��rEU�TVU�eCP�U�R a_�"YZ�sEWjadTp aV_J}onE\drjiNa�R noTdadrEU�T�Kie�Z�[ iNsEq�PNiNsEU-WjaV_JadroiNsocIe�\VU-U [K���Z�

Page 29: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� � � � ��� ������������� ����� � �FXaV_JR noTdadrEU�T�ZbWL�bZ�sEq-U-\�K � _bnoT �9U�Tp c7iNsEW iNs4aVU�TVU-\da�Z�sEW t9_�_�W ��i�\drEU-\ Z�TVU[ Z�aVU�Tdi�Z�P Zb\V\di�\dakZ�sEq-U aV_I[IU iNs0TVU�sEU ��iNsot�[K�U�sEU�Tdt#i�U-\~_bseZjaV_b}oi�q iNs��roi�qVr adrEU�TVU�r:Zb\ sE_ba Z�P � ZzL\W��U-U�sB[ nEqVr U�sEq-_bnoTkZ�t9U�[IU�s4a R _bT=[K��_bTdc Z�sEW iNs ��roi�qVr+e�r:ZbW�Z�PN[I_#\da�q-_b[IUCaV_fR U-U�P�adr:Z�a�e ��_bnoP�W sE_ba ��UZ�P�U�aV_fWo_g[gnEqkr [I_bTVU rEU�TVU�Z R aVU�T��i�onoa e"R U-U�P:WLi�"�U�TVU�s4adP�I\diNsEq-U TVU�ZbWLiNsot9_bnoT P�U�adaVU�TKS { �E}o}0�#v � �[�#v#mdT2K

��U�}:Zb\V\kZ�t9U�[I_bs4adTVU ���4nEU�P }�_biNs4afq-U�agp-qkr:Z�sot9U Z�U�nenoseiN[j}:Zbq�af}E\pLqVrE_bP�_bt#i��7nEUWop-q�i�\diSRC\dnoT�P�Z }�_bnoTV\dnoiNaVUYWoU-\�TVU-qVrEU�TVqkrEU-\ U�s p-q-_bP�_bt#i�UF� �7n:Z�sEW _bs \ Z�iNaIq-_b[K�oi�U�s]`_badcbZfU�a h+_bPNaVU�TdTkZf_bs4a�p�aVp�iN[j}�_bTdakZ�s4aV\ WEZ�sE\ P�U Wop��9U�P�_b}o}�U�[IU�s4a WoU�q-U�adaVU^\Vq�i�U�sEq-U#K

]+� p-qVrE_=TVU�sEWLn }:Z�T^P�Uf[I_bsEWoU \Vq�i�U�s7adi�):�4nEUjZ�nL�YadTkZ-�#Z�nL� WoUg]"_badcbZIsEUfR noa }:Zb\WoU-\ }oPNnE\=iNs4aVU�sE\VU soi~WoU-\ }oPNnE\=}�_#\ViNadiSR \�K���U�P�Z ��i�U�s4a�q-U�TdakZ�iNsEU�[IU�s4a�WLnBR Z�iNa �4nEU\V_bs PNiN��TVUF��\�� iNPQU���}oP�_bTVU U�s �7nEU�P��4nEU \V_bTdaVU�nosEU0sE_bno�9U�PNP�U0[ Z�soi���TVU W���Z���_bTVWoU�T=P�Z\Vq�i�U�sEq-UF��sEU�q-_bTdTVU-\d}�_bsEWg}:Zb\�Z�nL�g\VakZ�sEWEZ�TVWo\8WoU�q-U+[I_bsEWoU �7n`� iNP�Z�iN[IU�TkZ�iNa�}�_bnoTdakZ�s4aiNs4aVp�TVU-\V\VU�T�K9��U�P�ZC\VU�TVU-\V\VU�s7a���W�� nosEU�}:Z�Tda���P�_bTV\V�7nEU�TVp�}�_bsEWEZ�s7a Z�nL� U��Lq�nE\kU-\��7nEU R Z�iNah+_bPNaVU�TdTkZ }�_bnoTCsEUf}:Zb\JZ-�9_biNTCR�Z�iNaQ[IU�s7adi�_bs WoU-\[* 4Y6 � 6%;:G:, WEZ�sE\~\VU-\CWLi�"�p�TVU�s7aVU-\}onN�oPNi�q�Z�adi�_bsE\��4]`_badc#ZJq-_bsEq-�-WoUQ�4nEU \k_bsI_7U�no��TVU U-\da [ Z�P�q-_bsos4nEU^WEZ�sE\ P�U [jiNPNi�U�n=U�a��W���Z�noadTVUg}:Z�Tda���WEZ�sE\CP�ZjTVU-Wo_bsEWEZ�sEq-UIZ��9U-q P�Zb�7nEU�PNP�U ]`_badcbZ }:Z�TdP�U�W�� nosEU p��9U�s7adnEU�PNP�U}onN�oPNi�q�Z�adi�_bs WoU=\VU-\ TVp-\dnoPNakZ�aV\ WEZ�sE\ aVU�PNP�U _bn�aVU�PNP�U=TVU��7nEU#K�]"Z0q-_bTdTVU-\d}�_bsEWEZ�sEq-U+�P�Zb�4nEU�PNP�UJsE_bnE\QZ-�9_bsE\^U�n Zbq-q-�-\��oU�s4adTVU ]`_badcbZIU�a�h+_bPNaVU�TdTkZ�E�#Z�WoUIM�X��#m���M�X��LM#K3S { �}o}0�9��Xd ��#v#vdT

�C� U-\da PM� �4nEU~q-_b[j[IU�sEq-Ufq-U�PNP�U~�7noi�sE_bnE\^U-\da�q-_bsos7nEUfU�s4adTVU~h+_bPNaVU�TdTkZ�U�a _ U�Z�TdP�KO^nBWop&�onoa���iNPQU-\Va=U-\V\VU�s7adi�U�PNP�U�[IU�s4a��7nEU-\dadi�_bs WoU-\=adTkZ-�#Z�nL� WoU qVr:Zbq�nos U�a=W�� nosEUTVU�sEq-_bs4adTVU~U�s4adTVUCP�U-\�WoU�nL� rE_b[j[IU-\�Ko�C� U-\da h+_bPNaVU�TdTkZ �7noi �4[ Z�PNt#TVpC\V_bs � t9U~Z-�#Z�sEq-pF�R U�TkZ�P�U0�9_�4Z�t9U WoU _ Z�Tdi�\9��]"_bsEWLTVU-\�K e2PCp�akZ�iNa }oTVp���n �4n`��Z�}oTV�-\ nosEU WLi�\Vq�nE\V\di�_bsh+_bPNaVU�TdTkZIZb\V\di�\daVU�TkZ�iNa(� nosEUfq-_bsLR p�TVU�sEq-UgWoU _ U�Z�TdP"P�Uf�4n:Z�adTVUfsE_��9U�[K�oTVU=M�X��9��K

]`U-\ WoU�nL��rE_b[j[IU-\+q-_bPNP�Z���_bTV��TVU�s7a }�_bnoT�P�ZC}:Z�Tdnoadi�_bs3�9U�s�M�X��#v�#W���Z�Tdadi�q�P�U-\+WEZ�sE\P�U^}�p�Tdi�_7WLi��7nEU��!1�� -0; ].<E/47E����R _bsEWopC}:Z�T _ U�Z�TdP�KoO^}oTV�-\�q-U�P�Z��h+_bPNaVU�TdTkZ adTkZ�sE\ [jiNa�� _ U�Z�TdP�nos�}:Z�}oi�U�T WEZ�sE\ P�U-�7nEU�P �"\dnoT P�Z+�:Zb\VU=WoU-\gp-�7n:Z�adi�_bsE\ WoU=q-U WoU�Tdsoi�U�TU�s q�Z�P�q�noP^WoU-\j�#Z�Tdi�Z�adi�_bsE\���iNP^Wop�[I_bs4adTkZ P���Z�s:Z�P�_bt#i�U�U�s4adTVU�P�ZeW7s:Z�[ji��7nEUYWop�[I_� t#TkZ�}oroi��4nEU U�a^q-U�PNP�UgWoU-\Q\p�\VaV��[IU-\ [ Z�aVp�Tdi�U�P�\�K���q-Uf[I_b[IU�s4a �PM��@P�U-\^p-qVr:Z�sot9U-\~WoU# �7i�U�sosEU�s7aQ}oPNnE\Q}�U�TV\k_bsosEU�P�\QU�aQP�U-\ P�U�adadTVU-\Cq-_b[j}oTVU�sosEU�s4a���}:Z�TQU��oU�[j}oP�UF�@P�U-\ �9_�U�nL�WoUK)Es W���Z�sosEp-U WLn q-_bno}oP�U�h+_bPNaVU�TdTkZ�Z�n q-_bno}oP�U _ U�Z�TdP�K`O^iNsE\Vi ��Z�P�_bTV\g�4nEU P�Uj\VU�i ��U�dZ�s7�7i�U�TjM�X��LMF��nosEU P�U�adadTVUjWoU _ U�Z�TdP+\���ZbqkrE���9Uj\dnoTQnos R _bTd[IU�P O7iNsEq-U�TdP�D9_bnoTV\ � ��P�U[���[IU _ U�Z�TdP@aVU�Td[jiNsEUC\kZgq-_bTdTVU-\d}�_bsEWEZ�sEq-U~U�s M�X��FXf}:Z�T � iNadr ��U-\Va+TVU�t4Z�TVWo\�R&TV_b[rE_bnE\VU~aV_jrE_bnE\VU �kK

�Qs0�9_biNa~Wo_bsEq �7n`��Z�n 2WoU�PM� WoU-\CZ��bZ�sEq-p-U-\~\Vq�i�U�s7adi�):�4nEU-\��:nosEU q-_bPNP�Z���_bTkZ�adi�_bseU�aWoU-\ p-qkr:Z�sot9U-\jWoU�q-U a 7}�U [I_badiN�9U�s7ajP�U-\ t#TkZ�sEWo\jqVrEU�TVqVrEU�noTV\ �4nEU \V_bs4ajq-U-\ adTV_bi�\rE_b[j[IU-\^U�a�P�U-\ _bs7a aV_bnoa^\diN[j}oP�U�[IU�s7a^TkZ�}o}oTV_�qVrEp-\�S { �o}o}0�FX��d ������dT2K

Page 30: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

��� ��������� ������� � � � ��� � ����� � � ���������

Page 31: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� ����������� �

������ ��� � � �������

� �� ��� �� � � ���_ U�sEqkrE_bsE\IsE_bnE\W� }oTVp-\VU�s7a \dnoT�P�U-\�[I_�Wo��P�U-\I�4noi � Z�n��d_bnoTVW�� r4noi^U�sEq-_bTVUF� \V_bs4a

noadiNPNi�\Vp-\�}:Z�T P�U-\ \kq�i�U�s4adi�):�7nEU-\^U�a^U�sE\VU�iNt#sEp-\QZ�nL�Yp�adnEWLi�Z�s7aV\�K� ��� � � ��� ��� � �! ������Q_bnE\=Z�PNP�_bsE\=q-_b[j[IU�sEq-U�T=}:Z�T iNs4adTV_�WLnoiNTVU0WoU-\ [I_7Wo��P�U-\=q-_bs4adiNs7nE\=}�_bnoTInosEU

U-\d}��-q-U#K �Q_baV_bsE\ n<j�o P�Z }�_b}onoP�Z�adi�_bs Z�n aVU�[j}E\ a�K ]`U�akZ�nL� WoU0q�TV_bi�\k\kZ�sEq-U�U-\daZ�P�_bTV\ Wo_bsosEpJ}:Z�TKZ g gij k s:Z�i�\V\ Z�sEq-U-\ q [I_bTdaV\�� [jiNt#TkZ�adi�_bsZ�}o}�U�P�p-U5wpx&yN{F|~}���� �Y� ���F�N�z���� d{F|~}����:K ]"U�[IU�[K�oTVUYWoUYWLTV_biNaVU�TVU-\daVU adTV�-\�t9p�sEp�TkZ�P �\di/�oi�U�s �7nEU�WoU-\Jq�Zb\C}:Z�Tdadi�q�noPNi�U�TV\gWop-q-_bnoP�U�s4a WoU q-U�adaVU�p-�7n:Z�adi�_bs`K�]`U�}oPNnE\J\diN[j}oP�UU-\daCq-U�PNnoi+_�v iNP�s`� Z=}:Zb\JWoU [jiNt#TkZ�adi�_bs U�aJ_�v P�U-\~s:Z�i�\k\kZ�sEq-U-\JU�a~P�U-\~[I_bTdaV\J\V_bs4a}oTV_b}�_bTdadi�_bsosEU�PNP�U-\]� � Zg gij k�� q�� ��� n<jpo k �����������! � H { KNM J_�v � U�a � \V_bs4a�WoU-\ q-_bsE\dakZ�s4aVU-\�}�_#\diNadiN�9U-\[Q Z0TVU�}oTVp-\VU�s4aVU P�UIakZ�nL� WoUIs:Z�akZ�PNiNaVpU�a�� q-U�PNnoi�WoU�Wop-q-�-\!Q U�a" � �bZ�noa" n$#Yo K �~�-\�P�_bTV\���\di �&% � ��P�Z0}�_b}onoP�Z�adi�_bsq�TV_ � a U���}�_bsEU�s4adi�U�PNP�U�[IU�s7a���Z�P�_bTV\ �7nEU�\di �(' � �+U�PNP�U�\%� p�aVU�iNs4a�K��C� U-\Va P���Z�}o}oTV_�qVrEU[ Z�PNadr4nE\di�Z�soi�\daVUF�L�7noi@U-\da�}oPNnoa�$ba�}�U�n TVp�Z�PNi�\VaVUF�4[���[IUC\�� iNP�U���i�\daVUQWoU-\�U��oU�[j}oP�U-\��4noiP�Z�q-_bs)ETd[IU�s4a�K

h+U�Tdr4noP�\daj[I_�WLi�)@Z q-U�adaVUYp-�7n:Z�adi�_bs U�sBZ�}o}�_bTdakZ�s4ajnos p�P�p�[IU�s7aI�4noi PNiN[jiNaVU�P�Zq�TV_bi�\V\kZ�sEq-UJP�_bTV\V�7nEU~P�Z }�_b}onoP�Z�adi�_bs WoU���i�U�s4a adTV_b}�iN[j}�_bTdakZ�s4aVU Zg gYj k&) n+* q -,/. o H { K�� J

�LM

Page 32: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

��� ��������� ������� ��� ������� ����� � � ����� �

_�v ) U�a . \k_bs4a~WoU-\Qq-_bsE\dakZ�s7aVU-\~}�_#\ViNadiN�9U-\�K��C� U-\daQP��Mwpx&yN{F|~}�����������}?�#|~}�x&y%��K�]`UgakZ�nL�WoU s:Z�akZ�PNiNaVp �#Z�noa�i�q�i *uq -,/.^�#U�a�iNP7R�Z�noa sE_baVU�T+�7n`� iNPLU-\Va8R _bsEq�adi�_bsIWop-q�TV_bi�\V\kZ�s7aVU WoU K . TVU�}oTVp-\VU�s4aVUCP�U-\�[I_�9U�sE\�WoU~\dnN�E\Vi�\dakZ�sEq-UQWLi�\V}�_bsoi��oP�U-\ _bn3�oZ�noadTVU�[IU�s4a WLiNa���P�Zq�Z�}:Zbq�iNaVp~WoUQP�� U�s4��iNTV_bsosEU�[IU�s4a � \dno}o}�_bTdaVU�T�P�� U-\V}��-q-UCU�a�T��7P�Zg�7iNaVU-\V\kU �gP�Zb�4nEU�PNP�U~q-U\VU�noiNP�U-\daJZ�adaVU�iNs4a�K�]+� p-�7n:Z�adi�_bs { K��=ZbWL[IU�aJWoU�nL�0}�_biNs4aV\~W�� p-�7noiNPNi��oTVU�H g -, gij k # J�7noi`\V_bs4a k # U�a� k . K k # U-\da�iNsE\VakZ��oP�Ugq�Z�Tg gij k&) q )

. t

� ) � n<jpo k

��� � U�a iNP��Z q�TV_bi�\V\kZ�sEq-U�U��L}�_bsEU�s7adi�U�PNP�U ��}:Z�Tjq-_bs4adTVUF�� k .lU-\da\dakZ��oP�UF�:q�Z�T g gYj k ) q )

. t

k ) n . � o�q )

.n . �

o t� ) . � ) q )

.n . t �� . o� ) . � ) q ) . q � ) � q )

� n<jpo k n . � o � � � � U�a � aVU�sEW �9U�TV\". K�]+� p-�7n:Z�adi�_bs { K�� Z�}�_bnoTj\V_bPNnoadi�_bs

t9p�sEp�TkZ�P�U Z n<j�o k � . ��� �

. � �in � � � q *�o� ��� ����� n<j�o k .��

�Qs�adiNTVUIWoU { K����7nEUI\di �� % .^�"Z�P�_bTV\ n<jpo Wop-q�TV_ � aL� nE\V�4n`� U�s . �8Z�P�_bTV\g�7nEUI\di � ' .^� n<j�o q�TV_ � a�K�e P"U-\da iNs7aVp�TVU-\V\kZ�s4a~WoUfTVU�[ Z�TV�7nEU�T~�4nEUf\Vi � ' . ,����Z�P�_bTV\P�Zjq-_bnoTp��UJWop-q�TdiNaVUf}:Z�T� n<j�o U-\da�nosEUf\diNt#[I_���WoUWSNM#Mi�L}o} M# ��dT2KO7i@_bs q-_bsE\di�Wo��TVUF�4U�s t9p�sEp�TkZ�P �9�4nEU P�U�akZ�nL��W���Zbq-q�TV_bi�\k\VU�[IU�s4a W�� nosEU }�_b}onoP�Z�adi�_bsU-\da Wo_bsosEpf}:Z�T g gij k�� n o � H { K�� Jnos }oTV_��oP���[IU \VU�}�_#\kU+Z�_bs q-_bsE\di�Wo��TVUI�7nEU�P�UjakZ�nL� WoU�s:Z�i�\k\kZ�sEq-U Z�t#iNaJiNsE\dakZ�s7akZd sEp�[IU�s7a��8\kZ�sE\gaVU�soiNTgq-_b[j}oaVU WLn�aVU�[j}E\fsEp-q-U-\V\ Z�iNTVUI}�_bnoTg�4nEUIP�U-\fsE_bno�9U�Z�nL� �sEp-\Z�adaVU�iNt#sEU�s7aQP�� � t9UgZbWLnoPNaVU#K �Qs�}�U�noa TVU�[Ip-WLi�U�T^U�s�[I_�WLi�)@Z�s4aQP�� p-�4n:Z�adi�_bs { K��+Zg gij k�� n n<jpo � n<jrq��]opo �_�v � �4U-\da�P�U }:Z�TkZ�[I��adTVUQTVU�}oTVp-\VU�s4akZ�s7a�P�U aVU�[j}E\�sEp-q-U-\V\kZ�iNTVUQ}�_bnoT Z�TdTdiN�9U�T �f[ Z�adn TdiNaVp#K9] � p-�4n:Z�adi�_bs WLi�"�p�TVU�s4adi�U�PNP�UQWLn Wop�P�Z�i �#nosEU�U��LaVU�sE\di�_bs WoU P�� p-�7n:Z�adi�_bsIP�_bt#i�\dadi��7nEUF�U-\da WoUgq-U~aa�}�U�Z g gij k&) n<jpo��+* q n<j�q��Ro. � � H { K {7J

Page 33: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� � ��� � ��� ��� � � ������� �� ���

_�v ) � .�U�a � \V_bs4a WoU-\gq-_bsE\dakZ�s7aVU-\ }�_#\diNadiN�9U-\%KC� s�P�Z�q-_b[j}:Z�TkZ�s4ac� { K���8_bs�TVU# [ Z�TV�4nEU~�7nEUQP�� U#"�U�a TVp�t#noP�Z�aVU�noT Wop�}�U�sEWY\ViN[j}oP�U�[IU�s4a WoUCP�Z }�_b}onoP�Z�adi�_bs0Z�n�aVU�[j}E\j0q�� }oPNnoa�$baC�4n`��Z�nYaVU�[j}E\ j K

|^sg[I_7Wo��P�U }oPNnE\"}�U�TdadiNsEU�s4a��4nEU { K { �%WLnfR�Z�iNa8�7nEU P�� U#"�U�a8WoU�Wop�P�Z�i7\VU�[!�oP�U WoU��9_biNTWop-q-_bnoP�U�T^W�� nosEUJ[I_�9U�sosEU \dnoT P�U-\�}�_b}onoP�Z�adi�_bsE\ }:Zb\k\Vp-U-\ U-\da Wo_bsosEpJ}:Z�T!Z

g gij k&) n<j�o � *Lq *.�� �� ��� n<j0q���o n���o g�� � �

_�v � n���o U-\VaJnos }�_bi�Wo\g�4noi+Wop�aVU�Td[jiNsEU ���4nEU�P+}�_biNs4agP�Z�}�_b}onoP�Z�adi�_bs�}:Zb\V\Vp-U Z�U�nnos0iN[j}:Zbq�aJ\dnoT^P�U-\QTVU-\V_bnoTVq-U-\~WLi�\d}�_bsoi��oP�U-\�K�� s }oTkZ�adi��7nEUF� � n<jpo aVU�sEW0�9U�TV\��I}�_bnoTj ' # U�a j \dnP \ Z�[j[IU�s4a t#TkZ�sEW�K4�CU }oPNnE\�� � n<jpo Z�noTkZJq-U�TdakZ�iNsEU�[IU�s4a�nosI[ Z �LiN[gno[U�senos q-U�TdakZ�iNs � % # KCO7i+_bs \ViN[j}oPNi�):UIU�seTVU�[j}oP�Z�b�Z�s4a � n<jpo }:Z�TJP�Z=R _bsEq�adi�_bs�WoU�QiNTkZbq � n<j q��]o �EZ�P�_bTV\� �� � � n<j q�� q���o n���o g� k n<j0q �]o

U�a _bs�TVU�aV_b[K��Uf\dnoT { K { KO7no}o}�_#\k_bsE\��7nEU n<j�#o k . }�_bnoT�nos�q-U�TdakZ�iNs j� U�a �4nEU n<j:q �]o ' . }�_bnoTj ' j� KE�~�-\�P�_bTV\��o}:Z�T { K { �:_bs Z *Lq� � � ��� � % # U�a^Wo_bsEq�� � � % # �:W�� _�v0_bs0}�U�noaWop-WLnoiNTVUJ�4nEU n<jpo q�TV_ � a U�s j K� s j k j � � � n<j0q��]o k n<j o k .�U�a Wo_bsEq� � � k # K _ _bnoT j� � � ' j ' j t H _bs Z�}o}�U�PNP�U j t P�� iNsE\dakZ�s7aC_�v �bZ�noa(�jsE_bno�9U�Z�n. J � n<j q �]o % .�Wo_bsEq � � � ' # U�a n<j�o Wop-q�TV_ � a � nE\k�4n`� U�s j k j t � � q�Z�Tq-Ugs`� U-\daJ�7nEUgPM�=�7nEU n<j\q �]o k n<j t o k .uU�aC�7nEU P�� _bs Z�noTkZ�� � � k # K���U�adaVUrEU�noTdi�\dadi��4nEUQ[I_bs4adTVU~�7n`� nos=q-_b[j}�_bTdaVU�[IU�s4a _#\Vq�iNPNP�Z�aV_biNTVU~U-\da�}�_#\k\di��oP�U#KY� s=R�Z�iNa��9P�U-\\V_bPNnoadi�_bsE\ WoU { K { \k_bs4a�}�p�Tdi�_7WLi��7nEU-\^WEZ�sE\�nos0t#TkZ�sEWYsE_b[K�oTVU WoUJq�Zb\�K

�^U�[ Z�TV�7nE_bsE\�U�sEq-_bTVU �7n`� nosEU p-�4n:Z�adi�_bs WLn a 7}�U g -, gij k � n o sEU }�U�noaZ-�9_biNT=WoU0\V_bPNnoadi�_bs }�p�Tdi�_7WLi��7nEU#K O7i^q#� p�akZ�iNa=P�U�q�Zb\��+WLi�\V_bsE\ Z-�9U-qYnosEU0}�p�Tdi�_�WoU � �Z�P�_bTV\� n<j � �]o k n<j�o Z

g -, gij k � n o� � g gij�� t k � n o g gij

� � ��� �� � g gij�� t gij k � ��� �� � n o g gYj gYjk � � ��� � � � � n o g k #

q-U^�7noi@U-\da�iN[j}�_#\V\Vi��oP�UF�7q�Z�T n~g -, gij�o t sEU }�U�noa ��adTVUQi�WoU�s4adi��7nEU�[IU�s4a�s7noP�U�s7adTVU j U�aj � � SNM#MY�L}o}�M1�d kM%��T2K

Page 34: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� { ��������� ������� ��� ������� ����� � � ����� �

� ��� � ����� � � � � � ��� � ��� � �� #� ����� � ����C_bnE\^Z�PNP�_bsE\�[ Z�iNs7aVU�s:Z�s4aC}oTVp-\VU�s4aVU�T^P�U~adTkZ-�#Z�iNP"�4nEU~P�U~[ Z�adrEp�[ Z�adi�q�i�U�s iNakZ�PNi�U�s

hCiNaV_0h+_bPNaVU�TdTkZ }onN�oPNi�Z U�s M�X��LMF��WEZ�sE\ � 6��>E/;C,5,�132 4F- G ���%E02:.?6 �L-G ��� �L- �G:.��/136 AC6 4F- 4Y1:G>G:6 8%E/1C2 4i- /.<6�K��~Z�sE\ q-U�a _bno��TkZ�t9UF�8h+_bPNaVU�TdTkZ0p�adnEWLi�U=P�U-\�#Z�Tdi�Z�adi�_bsE\ WLn=sE_b[!�oTVUJW�� iNsEWLiN�7i�WLnE\ �7noi�q-_bsE\dadiNadnEU�s7a P�U-\�WLi�"�p�TVU�s4aVU-\�U-\V}��-q-U-\�W�� nos[jiNPNi�U�n0Wo_bsosEp#K��������� ���� ������������������������� ����� "!������

e s4aVp�TVU-\k\V_bsE\ �sE_bnE\ Z�nL� akZ�nL� WoUCq�TV_bi�\V\kZ�sEq-U~WoUCqVr:Zbq�nosEU~WoU-\ U-\d}��-q-U-\%KO7i ��WEZ�sE\P�Ujq�Zb\J_�v P�U-\~[I_�9U�sE\ WoU�\dnN�E\di�\dakZ�sEq-Uj\V_bs7aJiNPNPNiN[jiNaVp-\���P�U-\~}�_b}onoP�Z�adi�_bsE\f_bs4af}�_bnoTakZ�nL��WoUfq�TV_bi�\V\ Z�sEq-U ) U�a ) t �EU�a �7nEUJP�� noadiNPNi�\kZ�adi�_bs WoUfq-U-\�[I_�9U�sE\QWEZ�sE\^P�U~aVU�[j}E\U-\da�sE_baVp-U$# n � t o �oP�U-\�akZ�nL�YWoUfq�TV_bi�\V\ Z�sEq-Uf\V_bs4aCZ�P�_bTV\ Wo_bsosEp-\ }:Z�TKZ

) q % # n � t o � ) t q % t # n � t o �% U�a % t p�akZ�s4aCWoU-\ q-_bsE\dakZ�s4aVU-\Q}oTV_b}oTVU-\ ��qkr:Zbq�nosEUgWoU-\ U-\V}��-q-U-\�Ko�~�-\�P�_bTV\��E_bs0Zg gij k n ) \q % # n � t o U�a

g tgYj k n ) \q % t # n � t o t� � g '& gij q ) � *

% k � g '&� tgYj q ) t � *

% t� % t gun '& #o0q % gun (&� t o k n ) % t q ) t % #o gij� � '& � *)�+ n<jpo*)-,t n<jpo � � ��'. k0/ n ) % t q ) t % #o j21 ��'.� )�+ n<j�o

) ,t n<j�o[k )�+ n$#Yo ) ,t n$#Yo

� � �, )2+� �+3) , $� � � �'.

�QsYR�Z�iNa�P�� rY�}�_badrE�-\VUf�7nEU ) , % % ) t , % t U�a^_bs�_��oadi�U�s4a��� ����� *)�+*)-,t k �54��

��_b[j[IU P�U-\^[I_�9U�sE\JWoUg\dnN�E\Vi�\dakZ�sEq-U \k_bs4a^PNiN[jiNaVp-\��:sE_#\Q}�_b}onoP�Z�adi�_bsE\CsEU }�U�no�9U�s7a}oTV_bPNiSR p�TVU�T �CP�� iNs)Esoi�K#�~�-\ P�_bTV\��� U-\da���_bTdsEp�U�a+Wo_bsEq t QuU�s�R Z�iNa P�ZC}�_b}onoP�Z�adi�_bsWo_bs7a P�U�TkZ�}o}�_bTda ) , % U-\da P�U�}oPNnE\ }�U�adiNarQ Wo_biNa+aVU�sEWLTVU �9U�TV\^�~}�_bnoT �4nEU sE_badTVU WoU�T soi���TVU p-�7n:Z�adi�_bs=\k_biNa+\kZ�adi�\ R�Z�iNaVU#K#OB}:Z�TdadiNT�W�� nos q-U�TdakZ�iNsI[I_b[IU�s4a��7_bsI}�U�noa+sEp�t#PNiNt9U�TP�Z }�_b}onoP�Z�adi�_bs �7noi8\%� p�aVU�iNs4afU�a~_bs U-\da~WEZ�sE\CP�Z \ViNadn:Z�adi�_bs adTkZ�iNaVp-U�WEZ�sE\CP�Z=\VU-q�adi�_bs{ KNM�S � �L}o} Xd kM#M#T2K�������6� 78�������9 "!�������:<;=?>@��ACB(DE����>@�

�Qs q-_bsE\di�Wo��TVU WoU�nL��U-\d}��-q-U-\ Wo_bs4a P�� nosEU \kUIsE_bnoTdTdiNa�U��Lq�PNnE\diN�9U�[IU�s7a _bn�}:Z�T adi�U�PNP�U�[IU�s7aCWoUJP���Z�noadTVU#K �Qs0qVrE_bi�\ViNa^P�� iNsEWLi�q-U Mf}�_bnoT P�Zj�7i�q�adiN[IU U�a P�� iNsEWLi�q-U���}�_bnoT

Page 35: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� � � � ��� ��� � ��������� �� � � � � � ������� � ��� �� � �

0

10

20

30

40

y

1 2 3 4 5 6

x

� . ��� { KNM � t U�s�R _bsEq�adi�_bs0WoU P�U�}oTVp-WEZ�aVU�noT�K`]+� rY�}�_badrE�-\VUI\VU�P�_bseP�Zb�7nEU�PNP�UjP�U-\JakZ�nL�eWoUjq�TV_bi�\V\kZ�sEq-Uj\V_bs7agWo_bsosEp-\}:Z�T ) q % t � q ) t � % t �Z-�9U-q ) U�a q ) t P�U-\fakZ�nL� WoUIq�TV_bi�\k\kZ�sEq-UjWoU-\fU-\d}��-q-U-\f\di+U�PNP�U-\f\k_bs4aJi�\V_bP�p-U-\�� % U�a% t WoU�nL�0q-_bsE\dakZ�s4aVU-\C}�_#\diNadiN�9U-\��u�oi�U�s �7n`� nos0}�U�n \ViN[j}oPNi�\daVUF�@U-\Va^nos ��_bs }�_biNs7aCWoUWop�}:Z�Tda�K �Qs Z�Wo_bsEqg gij k �n ) \q % t o U�a

g tgYj k q t n ) t q % t o � H { K � J|Qs TkZ�i�\V_bsosEU�[IU�s4af}oPNnE\~TdiNt9_bnoTVU�nL�eq-_bs)ETd[IUjq-U-\JWoU�nL� p-�7n:Z�adi�_bsE\�KCO�_bi�U�s7afWoU�nL�U-\d}��-q-U-\�Wo_bs7a�P�U~akZ�nL� WoUJq�TV_bi�\V\ Z�sEq-UF�o\di�U�PNP�U-\�\V_bs4a�i�\V_bP�p-U-\��7�#Z�P�U�s4a�� U�a�� t K@O7no} }�_#\V_bsE\ �4n`� U�sInosEU^nosoiNaVp^WoU�aVU�[j}E\ iNP%IZ�iNa l t TVU�sEq-_bs4adTVU-\�� l % # U�a �7n`� U�PNP�U-\Z�i�U�s4a nosEUJiNs��EnEU�sEq-Ug}oTV_b}�_bTdadi�_bsosEU�PNP�U Z�nYaVU�[j}E\^\dnoT�P�ZIq�TV_bi�\V\kZ�sEq-UfWoU-\ U-\V}��-q-U-\[Zg k � �gij � l t s gij � g t k � t t gYj � l t s t gYj �Z-�9U-q s U�a s t P�U-\ q-_7U#P q�i�U�s7aV\^WoU~}oTV_b}�_bTdadi�_bsos:Z�PNiNaVp#K �QsYadiNTVUfP�UJ\p�\VaV��[IUJW�� p-�4n:Zd adi�_bsE\ WLi�"�p�TVU�s7adi�U�PNP�U-\!Zg gij k dn � � l s t o � g tgij k t n � t � l s t o � H { K�m J

Page 36: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

�#m ��������� ������� ��� ������� ����� � � ����� �

0

2

4

6

8

x, y

5 10 15 20

t

��. ��� { K�� � t U�s�R _bsEq�adi�_bs0WoU _ _bnoT���adTVU q-_brEp�TVU�s4ajZ-�9U-qIP�U-\grY�}�_badrE�-\kU-\g�7nEUjsE_bnE\ Z��9_bsE\fR�Z�iNaVU-\���sE_bnE\ WoU��9_bsE\}�_#\VU�T � % # � � t ' # U�a���q-_b[j[IU�P�U-\~TVU�sEq-_bs4adTVU-\ \V_bs4a~s7noi�\di��oP�U-\K�=P�Z=}oTVU�[ji���TVUU-\d}��-q-UfU�a�R�Z-�9_bTkZ��oP�U-\]��P�ZjWoU�nL�Li���[IUF� s ' # U�a s t % # K:h+_biNPM�j�7noiN� nE\Vadi�):UfWo_bsEqP�U-\ p-�7n:Z�adi�_bsE\^U�s { K � Ko�CU { K�m�o_bs�adiNTVUJ}:Z�T nos�q�Z�P�q�noP`}�U�n Z�adadTkZ�4Z�s4a

� + � ��� + , k�� � ,t � ��� , + � � k ��� q-_bsE\da�K|QsEUJZ�s:Z�P��\VUCt#TkZ�}oroi��7nEUfWoU~q-U�adaVUJp-�4n:Z�adi�_bs�WEZ�sE\ P�UC}oP�Z�s�H$ � t J S ���}o}eM � kM%��T[I_bs7adTVUj�4nEUF��\V_bnE\Cq-U�TdakZ�iNsEU-\Jq-_bsEWLiNadi�_bsE\���P�U-\Cq-_bnoTp��U-\~_��oaVU�s4nEU-\J\V_bs7aCR U�Td[Ip-U-\~U�a\�� U�s7�9U�P�_b}o}�U�s7aCZ�noaV_bnoT WLnY}�_biNs4a WoUJq-_�_bTVWo_bsosEp-U-\!H � +)2+ � � ,) , J Q �9_biNT�P�U-\�)Et#noTVU-\ { KNMU�a { K��LK��C� U-\da U�s�U#"�U�afnos�}�_biNs7ag\dakZ�adi�_bsos:Z�iNTVU q�Z�TgiNP�Z�sos7noP�UIP�U-\ Zbq-q�TV_bi�\V\VU�[IU�s4aV\WEZ�sE\ { K � K 8_bn�� _bnoTV\~WEZ�sE\C\VU-\ �\6��>E/;3,+,d132 4i- G ���%E02:.<6 �L-G � � �L- G:.��/136^A/64i- 4Y1:G>G:6 8%E0132 4i- /.?6:��h+_bPNaVU�TdTkZep�sE_bsEq-U�q-UY�7n`� iNP^Z�}o}�U�PNP�U�P�Z �C��} ��y � � �&� � w�� }�� ��}�x&y%� Z

]`U-\ �EnEq�adn:Z�adi�_bsE\^WoU-\^WoU�nL��U-\d}��-q-U-\�\V_bs7a�}�p�Tdi�_�WLi��4nEU-\%K _ _bnoT nos�q-U�T akZ�iNs q-_bno}oP�UIWoU��bZ�P�U�noTV\JWoU-\~sE_b[!�oTVU-\gW�� iNsEWLiN��i�WLnE\���P�� p�akZ�afWoU�P���Zb\V\k_� q�i�Z�adi�_bs5�oi�_bP�_bt#i��4nEU U-\daf\dakZ�adi�_bsos:Z�iNTVUIU�aJP�� p-�7noiNPNi��oTVUIU-\daf\dakZ��oP�U+S �N�@}`KM�XdT2K

Page 37: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� � � � ��� ��� � ��������� �� � � � � � ������� � ��� �� �9�

�Qs=adiNTVUCnos�TVp-\dnoPNakZ�a iN[j}�_bTdakZ�s7a WoUCP�Zg}�p�Tdi�_�WLi�q�iNaVpCWoU U�a t Z�_bs�TVp-p-q�TdiNa { K � Zg '& gij k&) q % t � g '& tgij k&) t q % t U�a _bs�iNs4aV��t#TVUg}�U�sEWEZ�s7a nosEUJ}�p�Tdi�_7WoU � Z

) � q % � �'. � ��'. t gYj k # � ) t � q % t � �'. � ��'. gij k # �q-UJ�4noi`p-�7noiN�#Z�noa �

) t% t k

*� � �'. � ��'. �gYj U�a ) % k *� � �'. � ��'. t gij �

O^iNsE\di ��P�U-\g�bZ�P�U�noTV\ \dakZ�adi�_bsos:Z�iNTVU-\ k �+)2+ U�a t k

�,) , \V_bs4agP�U-\f[I_�9U�sosEU-\ WoU-\�bZ�P�U�noTV\~}oTdi�\VU-\~}:Z�T U�a t }�U�sEWEZ�s7afP�Z=}�p�Tdi�_�WoU � K�h+_bPNaVU�TdTkZ�p�sE_bsEq-U Z�P�_bTV\~P�Z

�3��} �Y�K��{ � ���N�z��� d{i|~}~��� �Y�#���c� �i�&�%��� �WZ]`U-\J[I_�9U�sosEU-\���}�U�sEWEZ�s4a nosEU�}�p�Tdi�_�WoUF��WoU-\JsE_b[K�oTVU-\ WoU-\JiNsEWLiN��i�WLnE\WoU-\JWoU�nL� U-\V}��-q-U-\J\V_bs7aJiNsEWop�}�U�sEWEZ�s7aVU-\gWoU-\Jq-_bsEWLiNadi�_bsE\giNsoiNadi�Z�P�U-\���U�ap�t4Z�P�U-\ Z�nL� sE_b[!�oTVU-\ �4noi�q-_bTdTVU-\d}�_bsEWoU�s4a �gP�� p�akZ�a \dakZ�adi�_bsos:Z�iNTVUF�L}�_bnoTP�U-\ �#Z�P�U�noTV\ Wo_bsosEp-U-\ WoU-\ q-_�U#P q�i�U�s4aV\�W���Zbq-q�TV_bi�\V\VU�[IU�s7a � ) � ) t U�a+WoU-\ q-_�U#P q�i�U�s4aV\CWoU~�9_bTkZbq�iNaVp � % � % t S ��L}o} M�X�[���dT2K

O^}oTV�-\fnosEUjp�adnEWoUI�4nEU�P��7nEU�}�U�n R Zb\dadi�WLi�U�nE\kUF��h+_bPNaVU�TdTkZYp�sE_bsEq-UIU�sEq-_bTVUjP�Z �C��}�Y�K��{ ���&|~y ���p{F|~}���� �Y�#���c� �i�&�%��� ��Z

O7i:P�� _bs�Wop�adTdnoiNa�P�U-\�WoU�nL� U-\d}��-q-U-\+nosoiSR _bTd[Ip�[IU�s4a U�a�}oTV_b}�_bTdadi�_bsosEU�PNP�U# [IU�s4a�Z�nL�IsE_b[K�oTVU-\ WoU^P�U�noTV\ iNsEWLiN��i�WLnE\��7P�ZJ[I_�9U�sosEUCWLn sE_b[K�oTVUCWoU-\iNsEWLiN��i�WLnE\QWoU P�� U-\d}��-q-UgWop��9_bTVp-U q�TV_ � aCU�a~q-U�PNP�UgWoU P�� U-\d}��-q-UgWop��9_bTkZ�s4aVUWLiN[jiNs4nEU S � �o}`K��#mdT2K

�CU }oPNnE\��%\Vi9P�� _bs sEU�Wop�adTdnoiNa��7nEU+P�� U-\V}��-q-U�Wop��9_bTkZ�s4aVUF�b\kZ�[I_�9U�sosEU�TVU�P�Z�adiN�9U sEU �bZ�Tdi�U}:Zb\ [ Z�i�\jq-U�PNP�UYWoU P���Z�noadTVUYZ�not#[IU�s4aVU#K�O7i _bs sEU�Wop�adTdnoiNaI�7nEU=P�� U-\V}��-q-U�Wop��9_bTVp-UF�q#� U-\da+P�ZJ[I_�9U�sosEU^TVU�P�Z�adiN�9UQWoU P���Z�noadTVUQ�4noi:WLiN[jiNs7nEU#K� s)Es3�4\di@_bs }oTV_baV��t9U P�� U-\V}��-q-UWop��9_bTVp-UF�oP�U-\ WoU�nL�Y[I_�9U�sosEU-\~Z�not#[IU�s4aVU�s7a�K_ TVp-q�i�\V_bsE\ U�sEq-_bTVUF�:q-_b[j[IUJP�U~R�Z�iNa�h+_bPNaVU�TdTkZ WEZ�sE\�P�Z�q-_bsEq�PNnE\di�_bs0WoUJ\k_bsYPNiN�7TVUF��4n`� nos�[I_�Wo��P�UQsEU~aVU�s:Z�s7a�}:Zb\�q-_b[j}oaVUJWLnYWop�P�Z�i`U�s4adTVU~nosEU~q�Z�nE\VU~U�a \V_bs�U#"�U�a�U-\da[I_biNsE\Qq-_bsLR _bTd[IU��IP�ZITVp�Z�PNiNaVpF�@[ Z�i�\ �oi�U�s }oPNnE\C\diN[j}oP�UI�Ip�adnEWLi�U�TIS ��E}o} M { ��#m U�a��_bsEq�PNnE\di�_bsT2K���6����� �> � �@�������

�JZ�sE\� 0;>- � .��/1C69AC6%,R8@E/8%1C4F-G>.<E0;3,F� �^_bt9U�T �JZd� _�� R�Z�iNa�nosEU�q�TdiNadi��7nEU�\dnEq# q�iNsEq�aVUjWLn [I_�Wo��P�UjWoU�]`_badc#Zd �h+_bPNaVU�TdTkZLK���U�WoU�Tdsoi�U�TJadTkZ-�#Z�iNPNP�U Z-�9U-qjWoU-\CR _bsEq�adi�_bsE\q-_bs4adiNs4nEU-\���Z�P�_bTV\��7nEU P�U=sE_b[K�oTVU�W�� iNsEWLiN��i�WLnE\�W�� nosEU U-\V}��-q-U=U-\dagR _bTVq-p�[IU�s4a�nosEUR _bsEq�adi�_bs�WLi�\kq�TV��aVU#K"�~Uj}oPNnE\���P�UI[ Z�adrEp�[ Z�adi�q�i�U�s�iNakZ�PNi�U�s�sEUjadi�U�s4a q-_b[j}oaVUjsoi WoU-\t9p�sEp�TkZ�adi�_bsE\I�4noi _bs4aIWoU-\�q-_bsE\Vp-�7nEU�sEq-U-\j\dnoT�P�Z0R _bTd[IU�WoU-\ �#Z�Tdi�Z�adi�_bsE\���soi WoU P�Z

Page 38: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

�#v ��������� ������� ��� ������� ����� � � ����� �

sE_badi�_bs�WoU^Wop�P�Z�i�K � s=_bnoadTVUF�7P�U-\+TVU�P�Z�adi�_bsE\�}oTV_bi�U-\ �}oTVp-WEZ�aVU�noTV\ sEU^adi�U�sosEU�s4a�}:Zb\��7nEUWLn�r:Zb\kZ�TVW:�`aV_bnoa q-_b[j[IU P�Z R TVp-�7nEU�sEq-U WoUIP�U�noTV\ TVU�sEq-_bs4adTVU-\ s`� U-\da }:Zb\JR _bsEq�adi�_bsWoUJP�U�noTV\^\VU�noP�U-\^Z���_bsEWEZ�sEq-U-\^TVU-\d}�U-q�adiN�9U-\�KN� s)Es3�:iNP�s`� U-\da }:Zb\ R�Z�iNa�[IU�s4adi�_bs WoUJP�ZWop�}�U�sEWEZ�sEq-UgWLnYakZ�nL��WoU~s:Z�akZ�PNiNaVp!� P�ZIWoU�sE\diNaVpJWoUJ}�_b}onoP�Z�adi�_bs S�v�L}`K`M%�#��T2K

� � � ��������� � � � � ��� � ���OQ}oTV�-\ Z-�9_biNT�}oTVp-\VU�s4aVpCq-U-\ }�_biNs4aV\�WoU ��nEU adrEp-_bTdi��7nEU-\��4iNP:sE_bnE\�\VU�[K�oP�UQ}oTdiN[I_bT

WLi�Z�P`WoUJq-_bsLR TV_bs4aVU�TCq-U-\�q-_bsEq�PNnE\di�_bsE\ � P�Z TVp�Z�PNiNaVpJWoU~P�Z s:Z�adnoTVUfU�sY}oTVp-\kU�s4akZ�s4aCWoU-\_��E\VU�Td�#Z�adi�_bsE\^U�a WoU-\QU���}�p�Tdi�U�sEq-U-\ �7noi`_bs4aCp�aVp~[IU�sEp-U-\^WEZ�sE\^q-U]�onoa�K

|Qs }oTVU�[ji�U�T�U��oU�[j}oP�U U-\da=Wo_bsosEp }:Z�T P�U-\ �9p�t9p�akZ�nL��K ��i�U�s �7nEU�R _bsEWEZ�[IU�s akZ�P�U�[IU�s7a�WLi�"�p�TVU�s7aV\�WoU-\�U-\d}��-q-U-\YZ�soiN[ Z�P�U-\���iNP�\�\VU PNiN�7TVU�s7a9��nosEU q-_b[j}�p�adiNadi�_bs}�_bnoTIP�� U�Z�nBU�ajP�Z PNno[ji���TVU#K �CU-\ q�noPNadnoTVU-\ U��L}�p�TdiN[IU�s7akZ�P�U-\=WoUYadTV���:U-\I[I_bs4adTVU�s4a��}:Z�T^U��oU�[j}oP�UF�o�7nEU~P�Z�TVp-q-_bPNaVUJ}:Z�T�}oP�Z�s4aVU Wop-q�TV_ � a adTV�-\���iNaVU~P�_bTV\V�7nEU~P�Z�WoU�sE\diNaVpfWoUq-U-\ WoU�Tdsoi���TVU-\~Z�not#[IU�s4aVU#K_ Z�TQZ�iNPNP�U�noTV\��E}�_bnoT�WoU-\ WoU�sE\diNaVp-\^WLi�"�p�TVU�s7aVU-\��oP�Z TVp-q-_bPNaVU~}:Z�T�nosoiNaVpfWoUJ\dnoT R�Zbq-UaVU�sEW �9U�TV\JnosEU�[���[IU��#Z�P�U�noTK�=[IU-\dnoTVUj�7nEU P�U�aVU�[j}E\g\�� p-q-_bnoP�U#K �Qse}�U�noa~iNs7aVU�T }oTVp�aVU�Tfq-U TVp-\dnoPNakZ�aJt#T � q-Uc�=P�Z q-_b[j}�p�adiNadi�_bs }�_bnoT~P�Z=PNno[ji���TVU+Z@WoU-\~}oP�Z�s7aVU-\JadTV_b}sE_b[!�oTVU�nE\VU-\Q\kU~ti��sEU�s4a�K

]8Z�q-_b[j}�p�adiNadi�_bse}�_bnoT~P�� U�Z�n TVU-\k\V_bTda~[ji�U�nL� W�� nosEUjp�adnEWoU�\dnoT~P�U [ Z ��\�K��CU�nL�q�noPNadnoTVU-\IWoU�WoU�sE\diNaVp-\jWLi�"�p�TVU�s4aVU-\ _bs4aIp�aVp�_��E\VU�Td�9p-U-\+Z�nosEUYWoU�sE\diNaVp }oPNnE\jp�P�U��9p-U\���Zbq-q-_b[j}:Z�t#sEUYW�� nos sE_b[!�oTVUYW�� p�}oi�\�}:Z�T }oi�U-W [I_biNsEWLTVUYU�a�W�� nos TVU�sEWoU�[IU�s7a U�st#TkZ�iNs�}:Z�T^rEU-q�akZ�TVUf}oPNnE\ R�Z�i��oP�UF�E\kZ�sE\ �7nEU~P�Zjq-_bsE\V_b[j[ Z�adi�_bs W�� U�Z�n�sEUfqkr:Z�sot9UWS�v�}o}�M1� � �"M1�#mdT2K

��_bsEq-U�Tds:Z�s7a^P�U-\�U-\d}��-q-U-\ Z�soiN[ Z�P�U-\��LiNP`U��Li�\daVU~WoU-\�U��LU�[j}oP�U-\^WoUCq�Z�sosoi��:Z�PNi�\V[IU#K��rEU���P�U-\C}�U�TVqkrEU-\JWoUIO7i���p�Tdi�UF��P�U-\~\dn�� U�aV\CWoU t#TkZ�sEWoU�akZ�iNPNP�U \VU sE_bnoTdTdi�\V\VU�s7aK�Ir7noi� akZ�s7aVU }�_bnoT�q-U�s4a�WoU��dU�nosEU-\�WoU0P�Z [���[IU U-\V}��-q-U#K���U�P�Z }�U�Td[IU�a+� P�ZeR _bi�\=nosEUPNiN[jiNakZ�adi�_bseWLn sE_b[K�oTVUjW�� iNsEWLiN��i�WLnE\CU�aCP�Z=\dnoTd�7i�U WoUgP�� U-\V}��-q-U P�_bTV\V�7n`� iNP8s`� 0Z }:Zb\W���Z�noadTVU-\^}�_bi�\V\k_bsE\��LP�U-\��dU�nosEU-\�}�_bno�#Z�s4aQ\VU�noP�\ \VU~sE_bnoTdTdiNTCWoU~}oP�Z�sEq�aV_bs`K

��iNaV_bsE\���}oTVp-\kU�s4a��4nEU�P��7nEU-\"U��oU�[j}oP�U-\8W�� U-\V}��-q-U-\�Wo_bs4a8P�Z q-_bnoTp��U WoU q�TV_bi�\k\kZ�sEq-U\dnoiNa8nosEU�\ViNt#[I_���WoU#K�]8Z q�noPNadnoTVU WoU WoU�nL� Z�PNt#nEU-\�q�Z�P�q�iSR not9U-\�Z^[I_bs4adTVp�P���Zbq�adi�_bs�W�� nosR�Zbq�aVU�noT�PNiN[jiNakZ�s4a+P�ZQ��iNaVU-\V\VU�WoU�q�TV_bi�\k\kZ�sEq-U#KbO^n R noT�U�a �Q[IU-\dnoTVU WLn�aVU�[j}E\���P�U�akZ�nL�WoUIq�Z�P�q�iNno[ Z�not#[IU�s4aVU U�afP�ZYWLiN��i�\di�_bs�q-U�PNPNnoP�Z�iNTVU=TkZ�P�U�s4adiNa���q-UI�7noi Z�}�_bnoT q-_bsE\Vp# �7nEU�sEq-UJnosEUJWLiN[jiNs4noadi�_bs WoU~P�Z }�U�s7aVUfWoU~P�Z�q-_bnoTp��UJU�a�P�U~TVp-\dnoPNakZ�a^U-\da q�P�Z�iNTVU�[IU�s7aWLn�a 7}�UJ\diNt#[I_���WoU#K

]8Z q�TV_bi�\V\ Z�sEq-Ug\diNt#[I_���WoU }�U�noa(��adTVUg_��E\VU�Td�9p-U qVrEU�����U�Z�nEq-_bno} W�� U-\d}��-q-U-\!Z:P�U-\WLTV_#\V_b}oroiNP�U-\��:P�U-\CaVU�Td[jiNaVU-\��@P�U-\ R _bnoTd[ji�\��@P�U-\CZ���U�iNPNP�U-\��NKNKNK �Q_baV_bsE\C�7n`� U�sB)o�EZ�s4a~WoU-\

Page 39: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� � � � ������� ��� � ����� � ��� � �FXWo_bsosEp-U-\"U��L}�p�TdiN[IU�s7akZ�P�U-\8}:Z�Tdadi�q�noPNi���TVU-\��%iNP4U-\Va`}�_#\k\di��oP�U W�� _��oaVU�soiNT�WoU-\ �EnEq�adn:Z�adi�_bsE\}oPNnE\ _bn [I_biNsE\ iN[j}�_bTdakZ�s7aVU-\�WLn sE_b[K�oTVUeW�� iNsEWLiN��i�WLnE\�K _ Z�TYU��oU�[j}oP�UF��\VnoT=nosEUU-\d}��-q-U~WoUC[I_bnEqVrEUgWo_bs4a�P�U-\�R U�[IU�PNP�U-\ _bs4a q-U-q�i�WoU~}:Z�Tdadi�q�noPNi�U�T^�7nEUCP�U�noT�R p-q-_bsEWLiNaVps`� U-\da p�P�U��9p-Uf�7nEUJP�_bTV\V�7nEU~P�U~soiN�9U�Z�n WoU~P�Z�}�_b}onoP�Z�adi�_bs0U-\da �:Zb\�K

�C� U-\daJnosEU�iNPNPNnE\dadTkZ�adi�_bs WoU�P�� p-�4noiNPNi��oTVUIWoU-\f}�_b}onoP�Z�adi�_bsE\ Z�nos \p�\VaV��[IUjW7s:Zd [ji��4nEUIq�Z�}:Z��oP�U=WoUjP�U-\f[ Z�iNs7aVU�soiNTI�Ynos soiN�9U�Z�n�U�s�TkZ�}o}�_bTda Z��9U-qIP�U-\gTVU-\V\V_bnoTVq-U-\WLn [jiNPNi�U�n U�aI}�_bno�#Z�s4a Zb\k\dnoTVU�TIP���Zd� nE\daVU�[IU�s4a�WoU-\I}�_b}onoP�Z�adi�_bsE\ � WoUYsE_bno�9U�PNP�U-\q-_bsEWLiNadi�_bsE\�K

��U-\��7nEU�P��7nEU-\�U��oU�[j}oP�U-\ sE_bnE\�[I_bs4adTVU�s7a��oW�� _bTVU-\ U�a�Wop ����4�7nEU P�ZgadrEp-_bTdi�U^[ Z�P� adr4nE\di�Z�soi�\daVU sEU�}�U�noa \���Z�}o}oPNi��4nEU�T+Z�n�[I_bsEWoU �oi�_bP�_bt#i��7nEU�U�a��7n`� iNP�R Z�noa�\VU q-_bs4aVU�s7aVU�TWoU P�� iNs4aVU�Td}oTVU�aVU�T W�� nos }�_biNs4a WoU^��nEU i�Wop-_bP�_bt#i��7nEUIS�vN�9}o} M1�9� �@M#M1���M#M { ��M#M�v�@M1���dT2K

]+� p�adnEWoU^U��L}�p�TdiN[IU�s4akZ�P�UCWoU WoU�nL�IU-\d}��-q-U-\ U�s q-_b[j}�p�adiNadi�_bs [I��sEU W�� U�[!�oP�p-UCZ�nq-_bsEq-U�}oaCWoUgsoi�qVrEU�p-q-_bP�_bt#i��4nEUF��q#� U-\da �d 2WLiNTVU q-_b[j[IU�s4a���_�v U�aCZ�nL�0Wop�}�U�sE\QWoU �4noi\VU^sE_bnoTdTdiNa�aVU�PNP�UJU-\V}��-q-UF��}:Z�T��4noi�U-\da 2U�PNP�UC[ Z�sot9p-UF�Eq-_b[j[IU�s7a U�a�_�v�\VU^TVU�}oTV_�WLnoiNa U�PNP�UF�4\VU�TVU�}�_#\VU# �a 2U�PNP�U��]�+akZ�s4a Wo_bsosEp^�7nEU sE_bnE\�Z���_bTVWo_bsE\ U�s)Es P�Zf}:Z�Tdadi�U^�7nEU�P��4nEU}�U�nj}oTkZ�adi��7nEU^WoU�q-U�adTkZ��bZ�iNP �4sE_bnE\ s`��Z�PNP�_bsE\�}:Zb\ sE_bnE\ p�aVU�sEWLTVU�}oPNnE\ }oTV_�R _bsEWop�[IU�s7a\dnoT�P�Z�Wop#)EsoiNadi�_bs WoUfq-U�adaVUfsE_badi�_bs`K�Q_bnE\�Z�PNP�_bsE\"}oPNnoa�$ba�}oTVp-\VU�s7aVU�T�nosEU iNPNPNnE\dadTkZ�adi�_bs�WoU \V_bs iN[j}�_bTdakZ�sEq-U#K��JZ�sE\"nosEU

}oTVU�[ji���TVU U��L}�p�Tdi�U�sEq-UF� �fZ�nE\VU H M�XLM1���M�X#vFX J _��E\VU�Td�#Z WoU�nL� aa�}�U-\~WoU }:Z�TkZ�[Ip-q�i�U-\Zz4Z�s4aJP�Z=[���[IU soi�qVrEU#K�OQP�_bTV\~�4n`� U�seq�noPNadnoTVU�i�\V_bP�p-UF�@P�Z q-_bnoTp��U�WoU�q�TV_bi�\V\kZ�sEq-U�WoUqVr:Zbq�nosEU W�� U�PNP�U-\+Wop-q�TdiNa nosEU�\diNt#[I_���WoUF�9U�sIq�noPNadnoTVU�[jiS�LaVUF�#\VU�noP�U�q-U�PNP�U�Wo_bs4a P�U akZ�nL�WoUQq�TV_bi�\V\ Z�sEq-UCU-\Va�\dno}�p�Tdi�U�noT�\dnN�E\di�\daVUCU�a�P���Z�noadTVU~U-\da p�PNiN[jiNsEp-U~U�sYWoU�nL�=\VU�[ Z�iNsEU-\�K�~Z�sE\~nosEUIZ�noadTVU \Vp�Tdi�UjW�� U���}�p�Tdi�U�sEq-U-\���P�U-\fWoU�nL� }:Z�TkZ�[Ip-q�i�U-\g_bs7afWoU-\Jsoi�qVrEU-\g�4noiWLi�"���TVU�s7a �4nEU�P��7nEUY}�U�n3� P�� nosEUY��iNaj}oTV_7qkrEU0WoU-\c��_bTVWo\IWLn TVp-q�iN}oi�U�s7a��+akZ�sEWLi�\=�4nEUP���Z�noadTVUY}oTVp�R ��TVU�P�U�[jiNPNi�U�n WLn PNi��4noi�WoU#K+]/���P�U-\IWoU�nL� U-\V}��-q-U-\Iq-_br:Z��oiNaVU�s7a=U�ajnosp-�4noiNPNi��oTVUf\VakZ��oP�UJ\VU~[IU�a^U�s�}oP�Zbq-U�S�v�L}`K"M�m � T8U�a[S X�T2K

]+� U��L}�p�Tdi�U�sEq-UI\VnoiN�bZ�s7aVUIZ�p�aVp TVp�Z�PNi�\kp-UIU�s M�X�� � K`�~U�nL� aa�}�U-\fWoU�WLTV_#\V_b}oroiNP�U-\\V_bs4a [ji�\VU-\ U�sYq�noPNadnoTVUC[jiS��aVU~U�a���iNsEWop�}�U�sEWEZ�[j[IU�s7a^WoU-\�q-_bsEWLiNadi�_bsE\�iNsoiNadi�Z�P�U-\���nosp-�4noiNPNi��oTVU\)Es:Z�Po\�� p�akZ��oPNiNa�Kb|^sEU�WoU-\�WLTV_#\k_b}oroiNP�U-\ p�akZ�iNa�WoU�t#TkZ�sEWoU�akZ�iNPNP�UF���Qp��9_bPNnoadi�_bsP�U�s4aVU U�aI�YWLnoTVp-U WoUj�7i�U }oPNnE\fP�_bsot#nEU ��U�PNP�U=Z�adaVU�iNs7a q�iNsE�U��\ViS�eq-U�s7aV\giNsEWLiN��i�WLnE\��q-_bs4adTVUfWoU�nL�Y[jiNPNP�UJ}�_bnoT�P���Z�noadTVUWS�v�L}`K`M { ��T2Ke2P U��Li�\daVUjWoU-\fU��L}�p�Tdi�U�sEq-U-\f[I_bs4adTkZ�s4a��7n`� nosEUIZb\V\V_�q�i�Z�adi�_bs�\dakZ��oP�UjU�s4adTVU WoU�nL�U-\d}��-q-U-\��9p�t9p�akZ�P�U-\�Q \kZ�sE\��7n`� nosEU \V_biNa�p�PNiN[jiNsEp-U�}:Z�T�P�Z q-_b[j}�p�adiNadi�_bsmQ U-\da}�_#\V\Vi��oP�U#K:�~U-\Q\kU�[ji�\QWoU WoU�nL�0aa�}�U-\CWoUft#TkZ�[jiNsEp-U�\V_bs7aQ[Ip�P�Z�sot9p-\~WEZ�sE\CWLiN�9U�TV\VU-\}oTV_b}�_bTdadi�_bsE\+U�a+_bsj_��E\VU�Td�9UQZ�P�_bTV\ �4nEU P�_bTV\V�7nEU�nosjaa�}�U�WoU�t#TkZ�[jiNsEp-U U-\Va+U�s�U��oq-�-\��P�Z }oTV_b}�_bTdadi�_bs WoU~P���Z�noadTVUgZ�not#[IU�s4aVU U�a�iNs4�9U�TV\VU�[IU�s7a�S�v�L}`K"M � �dT2K

��iNaV_bsE\fU�sEq-_bTVU�P�� U��L}�p�Tdi�U�sEq-UI\VnoiN�bZ�s7aVUjq-_bsEq-U�Tds:Z�s4a WoU�nL� aa�}�U-\JWoU�PNiNs`K:e2sEWop# }�U�sEWEZ�[j[IU�s7a WoU-\g}oTV_b}�_bTdadi�_bsE\ WEZ�sE\fP�Uj[Ip�P�Z�sot9U WoUI\VU�[ji�\���P�Z�TVp-q-_bPNaVU _��oaVU�s4nEUZ-�9U-q P�Ug[Ip�P�Z�sot9U�U-\daC}oPNnE\CiN[j}�_bTdakZ�s7aVU �7nEU q-U�PNP�U _��oaVU�s7nEUIZ-�9U-q nos \VU�noP�aa�}�U WoUt#TkZ�iNsEU-\KS�v�L}`K`M � M#T2K

Page 40: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

{ � ��������� ������� ��� ������� ����� � � ����� �

]"_bTV\QWoUJP�� p�adnEWoU WoUfq-U�TdakZ�iNsEU-\CU-\d}��-q-U-\��EWoU-\ �#Z�Tdi�Z�adi�_bsE\^q&Lq�PNi��4nEU-\Q_bs4aCp�aVpf_�� \VU�Td�9p-U-\%K���rEU�� q-U�TdakZ�iNsE\=P�U�[j[jiNsot9\�U�a q-U�TdakZ�iNsE\=q�Z�[j}:Z�t#sE_bP�\���}:Z�T U��LU�[j}oP�UF��P�U-\q&Lq�P�U-\f_bs4agnosEU�P�_bsot#nEU�noTg[I_�9U�sosEU WoUI�4n:Z�adTVUIZ�sE\%K �Qs q-_bsE\dakZ�aVUIqVrEU��=q-U�TdakZ�iNsE\WoU�P�U�noTV\�}oTVp-WEZ�aVU�noTV\+q-_b[j[IU�P�U TVU�s:Z�TVWjZ�TVq�adi��7nEU�U�a�P�U�roi���_bn�WoU-\�sEU�iNt9U-\�WoU-\�q&Lq�P�U-\WoU�[���[IU�WLnoTVp-U#K _ _bnoT�W���Z�noadTVU-\�q-_bno}oP�U-\�}oTV_bi�U# �}oTVp-WEZ�aVU�noT��bP�Z WLnoTVp-U�WLn q&Lq�P�U }�U�noa�#Z�Tdi�U�T�K7��U�PNnoi@WLn P�7sL�IU�a WLnIPNi�����TVU WoU-\ sEU�iNt9U-\��#}:Z�T U��oU�[j}oP�UF�4\�� p�aVU�sEW \dnoT WLiS� Z�sE\S�v�o}o}0v#vd XLM#T2K

�~U-\ �oi�_bP�_bt#i�\daVU-\���Wo_bs4a �JZ�nE\VUF�@_bs4a~p�adnEWLi�p WoU-\~q-_bno}oP�U-\Q}oTV_bi�U# �}oTVp-WEZ�aVU�noTgZd)EsWoU=[IU�adadTVU=U�s p��7i�WoU�sEq-U q-U-\ _#\Vq�iNPNP�Z�adi�_bsE\�}�p�Tdi�_�WLi��7nEU-\�K"]"Z0}oTVU�[ji���TVU U���}�p�Tdi�U�sEq-UWoU �JZ�nE\VUgR noaCWoU sE_bnoTdTdiNT~nosEU }:Z�TkZ�[Ip-q�i�UIZ��9U-q nosEU P�U���noTVU!�oi�U�s }:Z�Tdadi�q�noPNi���TVU#K:e P_��E\VU�Td�#ZjWoU-\Qq&�q�P�U-\CZb\V\VU��JTVp�t#noPNi�U�TV\�ZoP�U-\ }:Z�TkZ�[Ip-q�i�U-\QTkZ�TVp#):U�s4aQP�Z�}�_b}onoP�Z�adi�_bs WoUP�Z�P�U���noTVUj}onoi�\ \VUjTkZ�TVp#):U�s4a U�PNP�U-\ �[���[IUF��Z�}oTV�-\g�7nE_bi+P�Z�P�U���noTVUj}�U�noa���sE_bno�9U�Z�n\���Zbq-q�TV_ � adTVUF�:P�U-\�}:Z�TkZ�[Ip-q�i�U-\ P�ZjTkZ�TVp#):U�s4a(� sE_bno�9U�Z�n U�a^Z�iNsE\di"WoUJ\dnoiNaVU

�fZ�nE\VUI)Eaf}oPNnE\di�U�noTV\ p�adnEWoU-\ Z��9U-qInosEU }:Z�TkZ�[Ip-q�i�U U�afnos�Z�noadTVU=q�iNPNi�p Wo_bs7agP�Z}:Z�TkZ�[Ip-q�i�U�U-\da P�ZY}oTV_bi�U#K��~Z�sE\ P�Z0}oTVU�[ji���TVU=WoU=q-U-\ p�adnEWoU-\��8iNP [IU�a�\diN[j}oP�U�[IU�s7aP�U-\JWoU�nL� U-\d}��-q-U-\~U�s }oTVp-\VU�sEq-U#K��JZ�sE\Jq-U�q-_bs4aVU��LaVUF��P�Z }oTV_bi�U�WLi�\d}:Z�TkZ � af\VnoiN�7i�U�WoU}�U�n�}:Z�T�P�U=}oTVp-WEZ�aVU�noT�K �JZ�sE\ P�Z WoU�nL�Li���[IU U��L}�p�Tdi�U�sEq-UF�8iNP�Zd�d_bnoaVU=nos \Vp-WLiN[IU�s7aWEZ�sE\ P�U-�7nEU�P�P�U-\ }:Z�TkZ�[Ip-q�i�U-\�}�U�no�9U�s4aI\VU TVp�R&not#i�U�TjU�aj\VU [gnoPNadiN}oPNi�U�T�K+]C���\VU�noP P�U}oTVp-WEZ�aVU�noT \%� p�aVU�iNs4a�K+�JZ�sE\�nosEUYWoU�Tdsoi���TVU�U��L}�p�Tdi�U�sEq-UF��iNP^Z�}o}�_bTdaVU�TVp�t#noPNi���TVU�[IU�s4aWoU-\C}oTVp-WEZ�aVU�noTV\JU�a��@PM���_bseZb\V\di�\VaVU!�=WoU-\ �EnEq�adn:Z�adi�_bsE\Jq&Lq�PNi��7nEU-\�S�v�@}o} M�m { kM�m#mdTU�aKS XdT2K

��U�adaVU adTV_bi�\di���[IU�U���}�p�Tdi�U�sEq-U U-\da����oi�U�s U�s7aVU�sEWLn3�#P�ZC}oPNnE\+q-_bsLR _bTd[IUL�CP�ZCTVp�Z�PNiNaVpWoUJP�Z s:Z�adnoTVU#Ku� PNP�UJiNPNPNnE\dadTVUfP�Z sEp-q-U-\V\diNaVpfWoU~P�� iN[j[jiNt#TkZ�adi�_bs }�p�Tdi�_�WLi��7nEUJWoUJ}oTVp-WEZd aVU�noTV\�}�_bnoT Z���_bnoadiNT �gnos TVp�t#iN[IUQq&Lq�PNi��4nEUCq-_bsLR _bTd[IU~Z�nL�I}oTVp���i�\di�_bsE\�adrEp-_bTdi��7nEU-\�K� sjOQnE\dadTkZ�PNi�UF�biNPEZQZ�iNsE\ViLp�aVp�}�_#\V\di��oP�U WoU q-_bs4adT�$bP�U�T P�ZQ}oTV_bPNiSR p�TkZ�adi�_bsIiNsEWop-\diNTkZ��oP�U�WoUq-U�TdakZ�iNsE\Cq�Zbq�adnE\ t#T � q-UK�IWoU-\ P�p�}oi�Wo_b}oaV��TVU-\�Ku� Z�i�\ q-_b[j[IU q-U�nL�7 2q�i8sEUf\�� iNsE\dakZ�PNP�U�s7a�7nEU^adTV�-\�P�U�s7aVU�[IU�s4a^WEZ�sE\ q-U�TdakZ�iNsEU-\ TVp�t#i�_bsE\��7P�U-\ q�Zbq�adnE\ ��p�sEp#):q�i���TVU�s4aQWoU^TVU�R not9U-\U�a s`� _bs7a }:Zb\ WLi�\d}:Z�Tdn S�vN�L}o} M�m { kM�m#mdT�U�aKS XdT2K

� ��� � � � � ���OQn aVU�Td[IUjWoUjq-U adTkZ-�#Z�iNP �`�4nEU�P��7nEU-\~}�_biNs4aV\J\V_bs7aK�=\k_bnoPNiNt#sEU�T�K "_bnoafW���Z���_bTVW:�

P�U-\ Z��bZ�sEq-p-U-\�Wo_bs7a�sE_bnE\ Z-�9_bsE\�}:Z�TdP�p~\�� iNsE\Vq�TdiN�9U�s7a q�P�Z�iNTVU�[IU�s4a^WEZ�sE\ P�Z [I_bno�#Z�sEq-UWoUgP�U�noT^p�}�_��7nEUF�o}oTVU�no�9UgU�s \V_bs4aQP�U-\ }oPNnE\di�U�noTV\Qq-_�� sEq�i�WoU�sEq-U-\Cq-_bsEq-U�Tds:Z�s7aQP�U-\QWop# q-_bno�9U�TdaVU-\+R Z�iNaVU-\ \diN[ noPNakZ�sEp�[IU�s4a�[ Z�i�\+WoU [ Z�soi���TVU iNsEWop�}�U�sEWEZ�s7aVU#KY� sE\VnoiNaVUF�#sE_bnE\sE_bnE\Q\V_b[j[IU-\ }�U�sEqVrEp-\Q\dnoT�P�U-\�}oPNnE\^q-p�P��&�oTVU-\QWoU-\^\Vq�i�U�s7adi�):�4nEU-\QWLn0Wo_b[ Z�iNsEUg�7nEUsE_bnE\jsE_bnE\=p�adi�_bsE\I)o�op � iNP U�s q-U�TdakZ�iNsEU�[IU�s7a ��U�Z�nEq-_bno}BW���Z�noadTVU-\=U�ajsE_bsBWoU-\[I_biNsEWLTVU-\�KC� s)Es3�"WEZ�sE\JP�Z�WoU�Tdsoi���TVU }:Z�Tdadi�UI\dnoTJP�U-\J[I_�Wo��P�U-\fU�aJP�U-\fU��L}�p�Tdi�U�sEq-U-\��sE_bnE\Js`��Z�noTdi�_bsE\ \dn5��adTVUIU��Lr:Z�nE\dadiSR \���sE_bnE\gZ-�9_bsE\ \Vp�P�U-q�adi�_bsosEp-\fP�U-\gp�P�p�[IU�s4aV\g�7noisE_bnE\�\VU�[!�oP�Z�i�U�s4a P�U-\ }oPNnE\ }:Z�TdP�Z�s7aV\�KNe P�\kU�TkZ�iNa \ Z�sE\ Z�nEq�nos�Wo_bnoaVU~adTV�-\ iNs7aVp�TVU-\V\kZ�s7aW�� p�adnEWLi�U�T~}oPNnE\CU�s0}oTV_�R _bsEWoU�noTCP�U-\Cq-_bs)ETd[ Z�adi�_bsE\~_bn3��Z�n q-_bs4adTkZ�iNTVUF��P�U-\QiNs)ETd[ Zd adi�_bsE\��4n`��Z�}o}�_bTdaVU�s4a8P�U-\8U��L}�p�Tdi�U�sEq-U-\"U�a"P�U-\8_��E\VU�Td�#Z�adi�_bsE\8Z�nL�f[I_�Wo��P�U-\`adrEp-_bTdi��7nEU-\�K

Page 41: users.skynet.beusers.skynet.be/fab_et_ed/blog/pd-lotka-volterra.pdf · ]`U-\ W s:Z [ji 7nEU-\ WoU } _b}onoP Z adi _bs=aVU sEWoU s7a Jp sE_bsEq-U T WoU-\+P _bi \ Wop-q TdiN bZ s7a

� ����� ����� ��� �������

SNM#T �I.?=�G:.?E0;3;>-0.?2:6 6%;3= 0=>47E/8��@AC.��/136 �c13.?474Y6NG0� ]�i��oTkZ�iNTdi�U OQTdi�\dadi�WoUVQnoiNPNP�U�a�� _ Z�Tdi�\��"M�X#m#vLKS��dTIO7r:Z�TV_bs � K �~iNsot9\dP�Z�sEW:���^E0AC6%47.?;���[-G:132:6>� rEU^|QsoiN�9U�TV\diNaaI_�R8��roi�q�Z�t9__ TVU-\V\��:��roi�q�Z�t9_N�8M�X#v � KS��dT �I.?=�G:.?E0;3;>-02 E�� ��=�.<6%;:G:. �.?= (.<E��02�-08 ��\����r:Z�TdP�U-\!O�q�Tdi��osEU�T�� \!O�_bsE\���QU ��� _bTdc>�`M�X#vLM#K

S { T �Ci�_bTdt#i�_ e \dTkZbU�PJU�a�OQs:Z^� iNPNP��s �fZb\Vq�Z� � �36 ].?E04YE��� E�����1��� u6@2:,F���iNTdc7r���nE\VU�T^h+U�TdP�Z�t%� � � P�UF�[�������LK

S � T O^PSR TVU-WeX4Z�[IU-\ ]`_badcbZ�0* 4Y6�� 6@;:G:,BE�� ���0,d.?=u-/4 (.<E04YE��� �`TVp-p-WLiNadi�_bs Z* 476 � 6%;:G:,�E���� -G �36 �L- G:.<=u-/4 ].?E04YE���\���C_%�9U�T _ nN�oPNi�q�Z�adi�_bsE\�� �QU �� _bTdc���M�X � mLK

S�mdT ]�nEW ��iNt��9_bs ��U�TdakZ�P�Z�s">i� � ���%E02:.<6 � �%; �%2�-/4Y6^A/6%,+, 0,�G�� � 6%,�� 8 ����,�.��/1C6��� �.?E04YE���.?6��u8%, �=��CE/47E���.<6��u,dE/=>.<E/47E���.<6��@8 �3.?47E/,dE/8 �3.?6:�9�CnosE_7W:�_ Z�Tdi�\��`M�X9�G�LKS ��TghQiNaV_ h+_bPNaVU�TdTkZ� � 6��>E/;C, ,�132 4F- G ���%E02:.?6 �L-G � � �L- G:.��/136 A/6 4F-4Y1:G>G:6 8%E0132m4i- /.?6:� �JZ�noadroi�U�T �hCiNPNP�Z�TV\��+M�X��LMF��TVp�iN[j}oTVU-\V\di�_bs Z �+WLiNadi�_bsE\

X4Zbq-�7nEU-\ �JZ��:Z�i�uO�q-U�Z�nL�>�"M�XFX��LKS�vdT �^_bt9U�Tf�JZd�d_��i� � 0;>- � .��/1C6 AC6@,98%E08%134F-G:.?E0;3,i�/� Zb\V\V_bs U�a ��i�UF� _ Z�Tdi�\��

M�X9� { KS XdT ��K `�K �JZ�nE\kUF��� �%2:. �.?=u-G>.<E0;3,�6��/8 �%2:. � 6%;3GF-/4Y6@,WAC6 4F- G ���%E02:.<6 � -��G ��� �L- G:.��/1C6 AC6 4F- 4Y1:G>G:6 8%E/1C2 4i- /.<6>� �QU�Td[ Z�sos U�aY��i�UF� _ Z�Tdi�\��

M�X�� � KSNM1�dT ��KC` K �fZ�nE\VUF� � �C6 �uG:2@1�� ��4Y6 �iE02 *��/.?,�G:6%;3=>6>���~_��9U�T _ nN�oPNi�q�Z�adi�_bsE\��

�QU ��� _bTdc>�`M�X9��M#KSNM#M#T XEKC� K(� noTdTkZzi��� - G �36�� -G:.?=u-/4 (.<E04YE����� ���I;!��;:G:2@E0AC13=�G:.<E/;��O7}oTdiNsot9U�T �h+U�TdP�Z�t%� �CU � � _bTdc>�[�������LK

{ M