laurentpujo-menjouet [email protected]
TRANSCRIPT
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Biomathematics 2Bifurcation in R
Laurent [email protected]
University Claude Bernard Lyon 1 – France
March 25, 2021
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Table of contents
Fish harvesting management
Introduction
Four types of bifurcation with one parameter
Hysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Fish harvesting management
Consider the fish population growth logistic model:
x ′(t ) = r x(t )(1− x
K
)−E x(t ) t ∈ I ⊂R,
with r and K positive fixed constants and E a positive varyingparameter. E corresponds to the harvesting management of thefish population (red tuna, anchovies or cod for instance).
Ï Determine the equilibria depending on E .Ï Determine their stability.Ï Denoting fE (x) = r x
(1− x
K
), the model is then written as
x ′ = fE (x).
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Fish harvesting management
Equilibria:an equilibrium x∗ satisfies x ′∗ = 0 that is fE (x∗) = 0 or
r x∗(1− x∗
K
)−E x∗ = 0 which is equivalent to x∗
(r (1− x∗
K)−E
)= 0
Ï it is easy to see that x∗ = 0 is always an equilibrium regardlessof the values of E .
Ï the other equilibrium is x∗ = K
(1− E
r
)given that 1− E
r> 0.
Ï Denote x∗0 = 0 and x∗
1 = K
(1− E
r
).
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Fish harvesting managementStability:x 7→ fE (x) is easily represented as a parabola with x∗
0 always a rootand x∗
1 changing its value as E varies.
Ï if E > r then x∗1 < 0 (not realistic), then x∗
1 is unstable and x∗0
is LAS,Ï if E = r then x∗
1 = x∗0 = 0 then x∗
1 = x∗0 is a negative shunt.
Ï if E < r then x∗1 > 0, then x∗
1 is LAS and x∗0 is unstable,
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Fish harvesting management
Summary in one figure:
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Fish harvesting managementSummary in one figure:If E is too large, that is too much harvesting, x∗
0 = 0 is LAS andthus the population will extinct,If E is below r , that is reasonable harvesting, x∗
1 is LAS and thusthe population will survive.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Table of contents
Fish harvesting management
Introduction
Four types of bifurcation with one parameter
Hysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
L. Pujo-Menjouet Biomathematics 3 - page 8/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Reminder: autonomous ode
Consider the autonomous ode of order one
x ′(t ) = f (x(t )), t ∈ I ⊂R,with f satisfying the conditions of the Cauchy-Lipschitz theorem(existence and uniqueness). For instance, f ∈C 1(R) is sufficient.
Ï Remember that solutions x of the autonomous ode aremonotonous.
Ï Remember that solutions of these ode can not overlap(consequence of the Cauchy-Lipschtiz).
Asymptotic behavior of the solutions of such equations is thusquite framed.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Bifurcation in R with one parameter: why?
The main interest resides then in inserting a varying parameterc ∈R (this is much closer to biological problems where parametersmay vary) such that we get a new range of variety in the sensethat new equilibria can appear, or some disappear, and stabilitymay also change. This is what we call the bifurcation theory. Abifurcation is crossed when the phase portrait changes.
The basic mathematical techniques you will need are available from thefollowing web site (menu "COURS", then "Théorie des systèmes
dynamiques"): http: // bmm. univ-lyon1. fr/
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Bifurcation in R with one parameter: how?
Consider now the autonomous ode of order one
x ′ = fc (x), and c ∈R,with f ∈C 1(R) (for simplification we note x = x(t )).
The objective is now to explore the number of equilibria and theirstability when c changes along R: c is then called the bifurcationparameter.A good example was the Allee effect with exploitation (seen inBiomathematics 1 in Fall). Exploitation E could be the bifurcationparameter.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Table of contents
Fish harvesting management
Introduction
Four types of bifurcation with one parameter
Hysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Detailed content
Four types of bifurcation with one parameterSaddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Saddle-node bifurcation
Consider the following equation
x ′ = x2 + c and c ∈R,with fc : x 7→ x2 + c ∈C 1(R).
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Saddle-node bifurcationEquilibria:
an equilibrium x∗ satisfies fc (x∗) = 0. That isx∗2 + c = 0 which is equivalent to x∗2 =−c.
Three cases appear:Ï if c > 0: there is no solution, so no equilibrium,Ï if c = 0: there is one equilibrium denoted x∗
0 = 0,Ï if c < 0: there are two equilibria x∗− =−p−c and x∗+ =p−c .
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Saddle-node bifurcationStability:there are two ways to study the asymptotic stability: either bycomputing the sign of f ′
c (x∗) or studying the graph of fc (seeBiomathematics 1).
Ï if c = 0 : x∗0 is a positive shunt,
Ï if c < 0 : x∗− =−p−c is LAS and x∗+ =p−c is unstable.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Saddle-node bifurcationBifurcation diagram:
Ï continuous line: x∗ =−p−c is the LAS branch,Ï dashed line: x∗ =p−c is the unstable branch,Ï there is a positive shunt when c = 0.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Saddle-node bifurcationBifurcation diagram:
Saddle-node bifurcation at c = 0.A saddle-node bifurcation occurs when two equilibria appear ordisappear.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
How do we read a bifurcation diagram ?A vertical line gives us the phase portrait for a given value of cwhich allows us to deduce the chronicles (in exercise).
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Detailed content
Four types of bifurcation with one parameterSaddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Transcritical bifurcation
Consider the following equation
x ′ = x2 + cx and c ∈R,with fc : x 7→ x2 + cx ∈C 1(R).
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Transcritical bifurcationEquilibria:
an equilibrium x∗ satisfies fc (x∗) = 0. That isx∗2 + cx = 0 which is equivalent to x∗ = 0 or x∗ =−c.
Ï Note that x∗0 = 0 is always an equilibrium regardless of the
values of c.Ï The second equilibrium exists also regardless the values of c,and we denote it x∗
1 =−c.Ï If c = 0, both equilibria overlap.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Transcritical bifurcationStability:
Ï if c < 0 : x∗0 is LAS, x∗
1 is unstable,Ï if c = 0 : x∗
0 = x∗1 is a positive shunt,
Ï if c > 0 : x∗1 is LAS, x∗
0 is unstable.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Transcritical bifurcationBifurcation diagram:
Ï if c < 0 continuous line: x∗ = 0 is the LAS branch, x∗ =−c isthe unstable branch,
Ï if c > 0 continuous line: x∗ =−c, is the LAS branch, x∗ = 0 isthe unstable branch.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Transcritical bifurcation
Bifurcation diagram:
Transcritical bifurcation at c = 0.A transcritical bifurcation occurs when we keep the samenumber of equilibria but their stability changes.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
How do we read a bifurcation diagram ?
A vertical line gives us the phase portrait for a given value c whichallows us to deduce the chronicles.
L. Pujo-Menjouet Biomathematics 3 - page 26/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Detailed content
Four types of bifurcation with one parameterSaddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Supercritical pitchfork bifurcation
Consider the following equation
x ′ = cx −x3 and c ∈R,with f ∈C 1(R) (for simplification we note x = x(t )).
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Supercritical pitchfork bifurcation
Equilibria:
an equilibrium x∗ satisfies fc (x∗) = 0. That is
cx −x∗3 = 0 which is equivalent to x∗ = 0 or c −x∗2 = 0.
Ï Note that x∗0 = 0 is always an equilibrium regardless of the
values of c.Ï The two other equilibria exist only if c ≥ 0, and we denotethem x∗
1 =−pc and x∗2 =p
c.Ï If c = 0, the three equilibria overlap.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Supercritical pitchfork bifurcationStability:
Ï if c < 0 : x∗0 is the only equilibrium and it is LAS,
Ï if c = 0 : x∗0 = x∗
1 = x∗2 and they are LAS,
Ï if c > 0 : x∗0 is unstable while x∗
1 and x∗2 are LAS.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Supercritical pitchfork bifurcationBifurcation diagram:
Supercritical pitchfork bifurcation at c = 0.A supercritical pitchfork bifurcation occurs when the bifurcationdiagram looks (locally) like a pitchfork with the external branchesbeing LAS.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
How do we read a bifurcation diagram ?
A vertical line gives us the phase portrait for a given value c whichallows us to deduce the chronicles.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Detailed content
Four types of bifurcation with one parameterSaddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Subcritical pitchfork bifurcation
Consider the following equation
x ′ = cx +x3 and c ∈R,with f ∈C 1(R) (for simplification we note x = x(t )).
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Subcritical pitchfork bifurcation
Equilibria:
an equilibrium x∗ satisfies fc (x∗) = 0. That is
cx +x∗3 = 0 which is equivalent to x∗ = 0 or c +x∗2 = 0.
Ï Note that x∗0 = 0 is always an equilibrium regardless of the
values of c.Ï The two other equilibria exist only if c ≤ 0, and we denotethem x∗
1 =−p−c and x∗2 =p−c.
Ï If c = 0, the three equilibria overlap.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Subcritical pitchfork bifurcationStability:
Ï if c > 0 : x∗0 is the only equilibrium and it is unstable,
Ï if c = 0 : x∗0 = x∗
1 = x∗2 and they are unstable,
Ï if c < 0 : x∗0 is LAS while x∗
1 and x∗2 are unstable.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Subcritical pitchfork bifurcationBifurcation diagram:
Subcritical pitchfork bifurcation at c = 0.A subcritical pitchfork bifurcation occurs when the bifurcationdiagram looks (locally) like a pitchfork with the external branchesbeing unstable.
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
How do we read a bifurcation diagram ?
A vertical line gives us the phase portrait for a given value c whichallows us to deduce the chronicles.
L. Pujo-Menjouet Biomathematics 3 - page 38/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Saddle-node bifurcationTranscritical bifurcationSupercritical pitchfork bifurcationSubcritical pitchfork bifurcation
Remarks:
Ï Supercritical pitchfork bifurcations are often called safe or softbecause stability always exist regardless of the values ofparameter c. And when crossing the bifurcation point, newsolutions appear or disappear with low amplitudes.
Ï Subcritical pitchfork bifurcations are often called dangerous orhard because instability may be the only option for theequilibria. When crossing the bifurcation point, solutions maytotally lose their stability with high amplitude. We call this acatastrophe in mathematics.
Ï In most biological problems, trajectories do not tend toinfinity. There always exists a stabilizing effect.
L. Pujo-Menjouet Biomathematics 3 - page 39/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Table of contents
Fish harvesting management
Introduction
Four types of bifurcation with one parameter
Hysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
L. Pujo-Menjouet Biomathematics 3 - page 40/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Hysteresis
Consider the following equation
x ′ = c +x −x3 and c ∈R,with f ∈C 1(R) (for simplification we note x = x(t )).
Ï Study the equilibria in function of parameter c,Ï Study their stability in function of parameter c,Ï Draw the bifurcation diagram and give an interpretation ofthe results.
L. Pujo-Menjouet Biomathematics 3 - page 41/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Hysteresis
L. Pujo-Menjouet Biomathematics 3 - page 42/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Hysteresis
L. Pujo-Menjouet Biomathematics 3 - page 43/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Hysteresis
L. Pujo-Menjouet Biomathematics 3 - page 44/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Table of contents
Fish harvesting management
Introduction
Four types of bifurcation with one parameter
Hysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
L. Pujo-Menjouet Biomathematics 3 - page 45/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
Bifurcation with two parameters: the cusp case
Consider the following equation
x ′ = c +d x −x3, c and d ∈R,with f ∈C 1(R) (for simplification we note x = x(t )).
Ï Study the equilibria in function of parameters c and d ,Ï Study their stability in function of parameter c and d ,Ï Draw the bifurcation diagram and give an interpretation ofthe results.
L. Pujo-Menjouet Biomathematics 3 - page 46/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
The cusp case
L. Pujo-Menjouet Biomathematics 3 - page 47/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
The cusp case
L. Pujo-Menjouet Biomathematics 3 - page 48/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
The cusp case
L. Pujo-Menjouet Biomathematics 3 - page 49/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
The cusp case
L. Pujo-Menjouet Biomathematics 3 - page 50/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
The cusp case
L. Pujo-Menjouet Biomathematics 3 - page 51/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
The cusp case
L. Pujo-Menjouet Biomathematics 3 - page 52/53
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Fish harvesting managementIntroduction
Four types of bifurcation with one parameterHysteresis (exercise)
Bifurcation with two parameters: cusp case (exercise)
The cusp case
L. Pujo-Menjouet Biomathematics 3 - page 53/53