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MBW:Predator-prey models with delay and prey harvesting

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This is a simplified summary presentation of a paper entitled Predator-prey models with delay and prey harvesting by Annik Martin and Shigui Ruan published in the Journal of Mathematical Biology, Vol. 43, 247-267 (2001) [1].

Predator-Prey Models with Delay and Prey Harvesting

Abstract

Three models on predator-prey with delay and harvesting will be presented, namely: Generalized Gause model with prey harvesting and delay in the specific growth term, Gerneralized Gause model with prey harvesting in the predator response function, and the Wangersky-Cunningham model with prey harvesting. Theorems, lemmas and proofs for the stability analysis of the general models are omitted. Specific examples given in the paper for each model will be provided. The dynamics of the model regarding the effects of adding delay on the system and changing the harvesting rates on prey population will be illustrated through numerical simulations.

Overview

Mathematics used

Delay Differential Equations and the corresponding solutions.

Type of Model

Population Model

Biological system studied

Any generalized system of the predator-prey relation. Further reading examples include Killer Whales and the Arthropod predator-prey model.

Background

Predator-prey systems have been studied enormously in the past decades due to its very rich dynamics and significant role in the management of renewable resources. An example of a predator-prey system is the arthropod system and can be found here. See also MBW:Pod-Specific Demography of Killer Whales [1]. In practice, harvesting is based on maximum sustainable yield [2]. Over-exploitation might lead to instability of the system or extinction of the resources. Adding a constant harvesting rate in the prey population changes the equilibrium values. On the other hand, incorporating a delay on the system might lead not only to instability and oscillations in the solutions but also switching of stabilities.

Prior to mathematically studying this type of a model with a time delay, one might want to familiarize with the Delay Differential Equation. http://en.wikipedia.org/wiki/Delay_differential_equation

The generalized Gause-type predator-prey model is given by

Aurelio eqn1.jpg

where Aurelio x.jpg and Aurelio y.jpg denote the prey and predator population densities at time Aurelio t.jpg, respectively. The function Aurelio g.jpg is the prey specific growth term , Aurelio p.jpg is the predator response term and Aurelio d.jpg is the death rate of the predators. A constant time delay Tau.jpg can be incorporated in the model in the following:

  1. Prey specific growth term Aurelio g.jpg ;
  2. Predator response term Aurelio p.jpg in the predator relation; and
  3. Interaction term Aurelio interact.jpg in the predator equation

The inclusion of the time delay Tau.jpg for the different possibilities above are given in equations (2), (3), and (4), respectively.

Aurelio eqn234.jpg

The time delay in system (2) is based on the assumption that in the absence of predators, the prey satisfies a Hutchinson's equation. In system (3), the time delay can represent a gestation period or reaction time of the predators. System (4) depicts the well-known Wangersky and Cunningham model, where the delay in the system assumes that the rate of change of the predators depends on the number of predators present at some previous time.

Mathematical Models and Simulations

MODEL 1

Generalized Gause Model with Prey Harvesting and Delay in the Prey Specific Growth

The model in consideration is given by

Aurelio model1.jpg

where Aurelio mu.jpg is a constant, Aurelio f.jpg is the specific growth rate of the prey in the absence of predators, Aurelio xh.jpg is the response function, Aurelio J.jpg is the minimum prey population required for the predator population to establish itself, and Aurelio H.jpg is the constant-rate harvesting of the prey species Aurelio x1.jpg. Also, Aurelio f0.jpg and Aurelio f.jpg is continuous and decreasing in Aurelio x1.jpg. The function Aurelio hx.jpg satisfies Aurelio cond1.jpg where Aurelio xhg.jpg is the response function. The delay Tau.jpg is a nonnegative constant denoting that in the absence of predators, prey's growth depends on its population density only after a fixed period of time.

Example

Consider the system

Aurelio example1.jpg

where the equilibrium is at Aurelio xy.jpg = (20,15).

Numerical Simulations

Case 1: Aurelio tau0.jpg

The figure on the left shows that the predator and prey populations spiral around the equilibrium point (20,15) and hence it is asymptotically stable. On the right is a plot of the predator and prey populations converging in to their equilibrium values Aurelio x2.jpg = 20 and Aurelio y2.jpg = 15, respectively.

Aurelio fig1.jpg Aurelio fig2.jpg

Below is a figure showing the effect of harvesting on the dynamics of the predator and prey populations for Aurelio H.jpg = 5 and Aurelio H.jpg = 10 with Aurelio x0.jpg = 40 and Aurelio y0.jpg = 16.

Aurelio fig3.jpg

Case 2: Aurelio tau1.jpg

The figure below shows a bifurcating periodic solution for Tau.jpg = 0.826.

Aurelio fig4.jpg

For the simulations below, Tau.jpg = 0.826. The left figure shows the oscillatory behavior of the prey and predator populations. The effect of the constant harvesting rate on the predator population is depicted in the figure on the left with Aurelio H.jpg = 10 and Aurelio H.jpg = 15. Aurelio fig5.jpg Aurelio fig6.jpg



MODEL 2

Generalized Gause Model with Prey Harvesting and Delay in the Predator Response Function

For this type of model, we consider the system

Aurelio model2.jpg

where Aurelio c.jpg > 0 is the rate of conversion of consumed prey to predator, Aurelio d.jpg > 0 is the death rate of the predator in the absence of the prey, Aurelio H.jpg is the constant rate of harvesting of the prey, and Aurelio f.jpg is the specific growth rate of the prey in the absence of the predators where Aurelio f0.jpg and Aurelio f.jpg is continuous and decreasing in Aurelio x1.jpg. The capture rate of prey per predator, that is the functional response is given by Aurelio xhg.jpg where Aurelio cond2.jpg. The delay Tau.jpg is a nonnegative constant.

Example

To illustrate the dynamics of the model, let's consider the system

Aurelio example2.jpg

with the equilibrium at Aurelio xy.jpg = (40, 12).

Numerical Simulations

Case 1: Aurelio tau0.jpg

It can be seen from the figure below that the equilibrium (40, 12) is a stale node. The predator and prey populations do not spiral around the equilibrium but still tend to the equilibrium point.

Aurelio fig7.jpg

Case 2: Aurelio tau1.jpg

The figure below on the left shows that for Tau.jpg = 7, the equilibrium point (40, 12) is a stable focus. The figure on the right tells the presence of a limit cycle for Tau.jpg = 9.

Aurelio fig8.jpg Aurelio fig9.jpg

For Tau.jpg = 9, it can be seen from the figure below that when Aurelio H.jpg = 10 both the predator and prey populations oscillate about the equilibrium values. Also, when Aurelio H.jpg = 15 both prey and predator populations converge to the equilibrium values.

Aurelio fig10..jpg

MODEL 3

Wangersky-Cunningham Model with Prey Harvesting

Consider the following system:

Aurelio model3.jpg

where Aurelio r1.jpg denotes the growth rate of the prey population, Aurelio r2.jpg is the death rate of the predator population, Aurelio b.jpg is the measure of effect of predation on Aurelio x1.jpg, Aurelio c.jpg is the measure of effect of predation on Aurelio y1.jpg, Aurelio H.jpg is the constant-rate harvesting of the prey.Also, Aurelio aKx.jpg where Aurelio Kx.jpg is a density-dependent term representing the limitation on the growth of the prey other than predation. As mentioned in the background section, the delay Tau.jpg, a nonnegative constant is based on the assumption that the rate of change of predators depends on the prey and predator populations present at some previous time.

Example

For purposes of numerical simulations, let's consider the system

Aurelio example3.jpg

which has an equilibrium at Aurelio xy.jpg = (5, 68/5).

Numerical Simulations

Case 1: Aurelio tau0.jpg

When the time delay Aurelio tau0.jpg, it can be seen from the figure below that the predator and prey populations spiral toward the equilibrium (5, 68/5), thus asymptotically stable. The initial values used are Aurelio x0.jpg = 2 and Aurelio y0.jpg = 10.

Aurelio fig11.jpg

Case 2: Aurelio tau1.jpg

Aurelio fig12.jpg Aurelio fig13.jpg

The figures above are simulations for Tau.jpg = 0.05. The left figure shows a bifurcating periodic solution and the one on the right shows the behavior of the predator population for different values of Aurelio H.jpg.

Results and Discussions

In this section, interpretations of the simulations for the specific examples in the above models are presented. It should be noted that there are some differences between the plots in the original paper and the version given here.

In model 1, when Aurelio tau0.jpg, the equilibrium point is asymptotically stable. It was also shown that prey and predator populations converge to their equilibrium values in finite time. Adding a constant harvesting of the prey changes the equilibrium value of Aurelio y2.jpg showing its dependence on Aurelio H.jpg. That is, the more prey is harvested, the lower the number of predators at the equilibrium, and the less prey is harvested, the higher the number of equilibrium value of predators. Moreover, as depicted on the third figure, the smaller the Aurelio H.jpg, the faster the prey and predator populations go to the equilibrium Aurelio xy.jpg. Incorporating a delay in the prey specific growth term gives a bifurcating periodic solution (e.g. Tau.jpg = 0.826). As shown in the simulations, the prey and predator populations oscillates around the equilibrium. Furthermore, varying the harvesting constant Aurelio H.jpg changes the equilibrium value of predator population. The larger the Aurelio H.jpg, the smaller the Aurelio y2.jpg, and the smaller the Aurelio H.jpg, the higher the value Aurelio y2.jpg is.

In model 2, when Aurelio tau0.jpg, the equilibrium is a stable node. The prey and predator populations do not spiral toward the equilibrium but still tend towards it. When a delay is present in the predator response function, the equilibrium point is a stable focus for Tau.jpg = 7 and existence of a limit cycle for Tau.jpg = 9, that is, a bifurcating periodic solution. As in model 1, the harvesting constant affects the dynamics of the system. Since, Aurelio y2.jpg depends on Aurelio H.jpg , as before, the more prey is harvested, the lower the equilibrium value of predators and vice versa. It should also be noted that the harvesting rate has a stabilizing effect on the equilibrium of the system as shown on the 10th figure above.

In model 3, when Aurelio tau0.jpg, the equilibrium point is asymptotically stable. Both prey and predator populations tend to their equilibrium values in finite time. When a delay is present in the system, there is a bifurcating periodic solution for Tau.jpg = 0.05. Moreover, increasing Aurelio H.jpg decreases Aurelio y2.jpg and decreasing Aurelio H.jpg increases Aurelio y2.jpg . However, changing the values of the harvesting constant does not affect the general behavior of the solutions (in contrast to the effect of varying Aurelio H.jpg in model 2 with delay).

If further reading on Predator-Prey Models is desired one can begin research with the original paper or by the references acknowledged at the bottom of the Scholarpedia page http://www.scholarpedia.org/article/Predator-prey_model

Citations

Many research studies and papers have cited the original of this paper. In particular, a research paper called On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay by S. Ruan cites this paper due to the rigorous investigation of the Delay Differential Equation with a discrete time delay. On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay was published by Mathematical Modeling of Natural Phenomena which is a respected international research journal. This paper studies solutions to nonlinear Delay Differential Equations which is an extension of the linear equations in this paper. Solution techniques vary as one moves from studying linear equations to nonlinear equations although methods given here are a strong basis and often have many similarities.

Matlab Code

This section provides the matlab codes to generate the above numerical simulations. For given example of the generalized Gause model with prey and delay in the prey specific growth, here is the code example1.m . Plots for the example of the generalized Gause model with prey harvesting and delay in the predator prey function can be obtained using example2.m. Simulations of the specific example of Wangersky-Cunningham Model with Harvesting are provided by example3.m.

Further Reading

The Dynamics of Arthropod Predator-Prey Systems

References

  1. Martin, A. and Ruan, S. Predator-prey models with delay and prey harvesting , Journal of Mathematical Biology, Vol. 43, 247-267, August 2001
  2. wikipedia site on maximum sustainable yield