Interspecific strategic effects Interspecific strategic effects - - PowerPoint PPT Presentation

Interspecific strategic effects Interspecific strategic effects Interspecific strategic effects Interspecific strategic effects

• f mobility in predator
• f mobility in predator-
• prey systems

prey systems

F i X Fei Xu Joint work with Dr. Ross Cressman d D Vl i il K i and Dr. Vlastimil Krivan Department of Mathematics Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5

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Outline Outline

• 1. Introduction
• 2. The Model
• 3. Lotka-Volterra Model
• 4. The Rosenzweig-MacArthur model
• 5. Conclusion

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Introduction Introduction

We assume both prey and predators adjust their activities to We assume both prey and predators adjust their activities to maximize their per capita growth rates. Prey Predators mobile strategy mobile (active) strategy sessile strategy sessile (ambush)

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Introduction Introduction

The Model The Model

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Introduction Introduction

Foraging efficiencies of the mobile predator chasing g g p g mobile prey F i ffi i i f h bil d h i Foraging efficiencies of the mobile predator chasing sessile prey Foraging efficiencies of a sessile predator catching mobile prey

• b e p ey

Foraging efficiencies of a sessile predator catching

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sessile prey

Introduction Introduction

Table 1. Predator foraging efficiency.

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Introduction Introduction

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The Model The Model

Both prey and predators use game theory-based strategies to maximize their i l i h per capita population growth rates. Replicator equation: Replicator equation:

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The Model The Model

Both prey and predators use game theory-based strategies to maximize their i l i h per capita population growth rates. Smoothed best response: Smoothed best response:

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The Lotka The Lotka-

• Volterra Model

Volterra Model

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The Lotka The Lotka-

• Volterra Model

Volterra Model

The effect of a dominated strategy The effect of a dominated strategy Suppose a mobile predator has higher foraging efficiency than a sessile predator independent of the strategy of the prey The proportion of mobile predators is increasing

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and will evolve to 1 unless x evolves to 0.

The Lotka The Lotka-

• Volterra Model

Volterra Model

• r

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The Lotka The Lotka-

• Volterra Model

Volterra Model

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The Lotka The Lotka-

• Volterra Model

Volterra Model

The system has an interior equilibrium if and only if The interior equilibrium is globally asymptotically stable if it exists; The interior equilibrium is globally asymptotically stable if it exists;

• therwise,

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The Lotka The Lotka-

• Volterra Model

Volterra Model

Replicator equation

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Replicator equation

The Lotka The Lotka-

• Volterra Model

Volterra Model

Replicator equation

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The Lotka The Lotka-

• Volterra Model

Volterra Model

Using the smoothed best response dynamics, we have

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The Lotka The Lotka-

• Volterra Model

Volterra Model

Smoothed best response strategy dynamics

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The Lotka The Lotka-

• Volterra Model

Volterra Model

Smoothed best response strategy dynamics

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The Lotka The Lotka-

• Volterra Model

Volterra Model

No dominated strategy No dominated strategy gy gy We assume that The predator forages more efficiently if it has the opposite strategy as its prey. Prey are better able to avoid predation if they have the same strategy as the Prey are better able to avoid predation if they have the same strategy as the predator.

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Introduction Introduction

Nash equilibrium for the frequency dependent evolutionary game Nash equilibrium for the frequency-dependent evolutionary game between predators and prey is given by

(No individual can increase its fitness by altering its strategy.)

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Introduction Introduction

With the mobile proportions fixed at their nash equilibrium values, the population dynamics becomes where

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Introduction Introduction

The equilibrium of most interest now is one where both strategic behaviors are present for each species where

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The Lotka The Lotka-

• Volterra Model

Volterra Model

Trajectories of the LV system with strategy dynamics given by the replicator equation when E1 exists.

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The Lotka The Lotka-

• Volterra Model

Volterra Model

Trajectories of the LV system with strategy dynamics given by the replicator equation when E1 exists

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equation when E1 exists.

The Lotka The Lotka-

• Volterra Model

Volterra Model

Trajectories of the LV system with strategy dynamics given by the smoothed best response equation when E1 exists.

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The Lotka The Lotka-

• Volterra Model

Volterra Model

Trajectories of the LV system with strategy dynamics given by the smoothed best response equation when E1 exists.

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The Lotka The Lotka-

• Volterra Model

Volterra Model

The coupled system may also have equilibria where both predators and prey are present but the populations adopt pure strategies. are present but the populations adopt pure strategies.

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are unstable if they exist.

The Lotka The Lotka-

• Volterra Model

Volterra Model

The only other equilibria of the coupled system are when the predator population is extinct, and the prey population is at carrying capacity. Finally there is also the trivial equilibrium with no prey and predators Finally, there is also the trivial equilibrium with no prey and predators

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The Lotka The Lotka-

• Volterra Model

Volterra Model

An alternative Lotka An alternative Lotka-

• Volterra model and the global stability of E

Volterra model and the global stability of E1

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The Lotka The Lotka-

• Volterra Model

Volterra Model

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The Lotka The Lotka-

• Volterra Model

Volterra Model

The global asymptotic stability of the system can the be shown by The global asymptotic stability of the system can the be shown by considering the following Lyapunov function

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The Lotka The Lotka-

• Volterra Model

Volterra Model

The derivative of V is obtained as The trajectory must converge to an invariant subset of It can be shown that E1is globally asymptotically stable since all these

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It can be shown that E1 is globally asymptotically stable since all these trajectories converge to E1.

The Lotka The Lotka-

• Volterra Model

Volterra Model

Let M be the maximal invariant subset

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The Lotka The Lotka-

• Volterra Model

Volterra Model

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The Lotka The Lotka-

• Volterra Model

Volterra Model

Thus, every trajectory that is initially in the interior of M converges to E1.

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y j y y g

The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

We assume Th th l ti d it d t t d i b Then the population density and strategy dynamics becomes

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

where

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

Moreover, with the mobile proportions fixed at the equilibrium values, the population dynamics becomes

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

The system has three equilibria where the predator population goes to extinction and the prey population reaches carrying capacity to extinction and the prey population reaches carrying capacity.

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

T j i f h RM i h d i i b h li Trajectories of the RM system with strategy dynamics given by the replicator equation when Eh1 exists.

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

Trajectories of the RM system with strategy dynamics given by the replicator equation when Eh1 exists.

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

Trajectories of the RM system with strategy dynamics given by the replicator ti h E i t equation when Eh1 exists.

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

Interestingly, the situation changes when the strategy dynamics is given by the smoothed best response: the smoothed best response:

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

T j i f h RM i h d i i b Trajectories of the RM system with strategy dynamics given by the smoothed best response when Eh1 exists.

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The Rosenzweig The Rosenzweig-

• MacArthur model

MacArthur model

Trajectories of the RM system with strategy dynamics given by the smoothed best response when Eh1 exists.

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Conclusions Conclusions

We investigate the dynamics of a predator-prey system with the assumption W g y p p y y p that both prey and predators use game theory-based strategies to maximize their per capita population growth rates. The predators adjust their strategies in order to catch more prey per unit time, while the prey, on the other hand, adjust their reactions to minimize the chances of being caught chances of being caught.

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Conclusions Conclusions

Numerical simulation results indicate that, for some parameter values, the system has chaotic behavior. Our investigation reveals the relationship between the game theory-based reactions of prey and predators, and their population changes.

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Thank you! Thank you!

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