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Dynamical System of a Two Dimensional Stoichiometric Discrete Producer-Grazer
Model : Chaotic, Extinction and Noise Effects
Yun KangWork with Professor Yang Kuang and
Professor Ying-chen Lai ,Supported by Professor Carlos Castillo-Chavez
(MTBI) and Professor Tom Banks (SAMSI)
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Outline of Today’s Talk Introduce LKE – model, and its corresponding discrete
case; Mathematical Analysis: bifurcation study Biological Meaning of Bifurcation Diagram; Chaotic behavior and Extinction of grazer; Nature of Carry Capacity K and Growth Rate b, and
their fluctuation by environments: adding noise Interesting Phenomenal by adding noise: promote
diversity of nature Conclusion and Future Work
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Stoichiometry It refers to patterns of mass balance in chemical
conversions of different types of matter, which often have definite compositions
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most important thing about stoichiometry
we can not combine things in arbitary proportions; e.g., we can’t change the proportion of water and dioxygen produced as a result of making glucose.
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Energy flow and Element cycling are two fundamental and unifying principles in ecosystem theory
Using stoichiometric principles, Kuang’s research group construct a two-dimensional Lotka–Volterra type model, we call it LKE-model for short
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Assumptions of LKE Model
Assumption One: Total mass of phosphorus in the
entire system is closed, P (mg P /l) Assumption Two: Phosphorus to carbon ratio (P:C) in the
plant varies, but it never falls below a minimum q (mg P/mg C); the grazer maintains a constant P:C ratio, denoted by (mg P/mg C)
Assumption Three: All phosphorus in the system is divided into two pools: phosphorus in the plant and phosphorus in the
grazer.
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Continuous Model
p is the density of plant (in milligrams of carbon per liter, mg C/l); g is the density of grazer (mg C/l); b is the intrinsic growth rate of plant (day−1); d is the specific loss rate of herbivore that includes metabolic losses (respiration)
and death (day−1); e is a constant production efficiency (yield constant); K is the plant’s constant carrying capacity that depends on some external factors
such as light intensity; f(p) is the herbivore’s ingestion rate, which may be a Holling type II functional
response.
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Biological Meaning of Minimum Functions
),min(q
gPK
)/)(
,1min( pgP
e
K controls energy flow and (P − y)/q is the carrying capacity of the plant determined by phosphorus availability;
e is the grazer’s yield constant, which measures the conversion rate of ingested plant into its own biomass when the plants are P rich ( ); If the plants are P poor ( ),
then the conversion rate suffers a reduction.
pgP /)(
1
pgP /)(
1
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Continuous Case:b=1.2 and b=2.9
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Discrete Model From Continuous One
Motivation: Data collect from discrete time, e.g., interval for collecting data is a year.
Biological Meaning of Parameters : Modeling the dynamics of populations with non-overlapping generations is based on appropriate modifications of models with overlapping generations.
Choose
pa
cppf
)(
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Mathematical Analysis
We study the local stability of interior equilibrium E*=(x*,y*)
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Bifurcation Diagram and Its Biological Meaning
For continuous case: K=1.5
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Bifurcation Diagrams on Parameter b
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Bifurcation Diagrams on Parameter b
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Bifurcation Diagrams on Parameter K
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Relationship Between K and b:
From these figures, we can see that there is nonlinear relationship between K and b which effect the population of plant and grazer:For bifurcation of K, increasing the value of b,
the diagram of b seems shrink.For bifurcation of b, increasing the value of K,
bifurcation diagram seems move to the left
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Extinction of Grazer
From bifurcation diagram, we can see that for some range of K and b, grazer goes to extinct. What are the reasons?
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Basin Boundary For Extinction
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Global Stability Conjecture
We know that Discrete Rick Model :
x(n+1)=x(n)exp{b(1-x(n)/K)} has global stability for b<2, does our system also has this properties
More general, if we have u(n+1)=u(n)exp{f(u(n),0} with global stability, then the following discrete system:
x(n+1)=x(n)exp{f(x(n),y(n))+g(x(n),y(n))}, of g(x,y) goes to zero as y tending to zero, in which condition has global stability
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Nature of K and b
K is carrying capacity of plant, and it is usually limited by the intensity of light and space. Since K is easily affected by the environment, it will not be always a constant ;
b is maximum growth rate of plant, it will fluctuates because of environment changing.
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Adding Noise
Because of the nature of biological meaning of K and b, it makes perfect sense to think these parameters as a random number.
We let K=K0+ w*N(0,1)
b=b0+w*N(0,1)
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Then Most Interesting thing on parameter K :
Prevent extinction of grazer :
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Time Windows
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Scaling
Define the degree of existence :
R=average population of graze/ average population of plant
Then try different amplititute of noise w, then do the log-log scaling, it follows the scaling law.
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Future Work
We would like to use “snapshot” method to see how noise effects the population of grazer and producer;
Try to different noise, e.g. color noise, to see how the ‘color’ effect the extinction of the grazer;
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