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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.

## Categorical Summary

Predator-Prey models generally describe the relationship across trophic levels; these particular forms of predator-prey models add to the classic Lotka-Volterra model by introducing both a delay and additional prey harvesting. The purpose of these additions is to determine maximum sustainable yield, attempting to mitigate population catastrophes across different trophic levels. Similar propositions are discussed in MBW:Effect Of Seasonal Growth On Delayed Prey-Predator Model [3] and The Dynamics of Arthropod Predator-Prey Systems [4]. Both of these project focus on the specifics behind Lotka-Volterra and present a solid foundation for building on the added complications of prey harvesting.

## Overview

This model can be applied to any generalized biological system between predators and their prey. The mathematics used in this model were delay differential equations and their corresponding solutions. The type of model employed in this page is a population model. Further reading examples include the Pod-Specific Demography of Killer Whales [1]and Arthropod predator-prey model[5] pages.

## 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 [3] is given by

where and denote the prey and predator population densities at time , respectively. The function is the prey specific growth term , is the predator response term and is the death rate of the predators. A constant time delay can be incorporated in the model in the following:

1. Prey specific growth term  ;
2. Predator response term in the predator relation; and
3. Interaction term in the predator equation

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

The time delay in system (2) is based on the assumption that in the absence of predators, the prey satisfies a Hutchinson's equation [4]. 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

where is a constant, is the specific growth rate of the prey in the absence of predators, is the response function, is the minimum prey population required for the predator population to establish itself, and is the constant-rate harvesting of the prey species . Also, and is continuous and decreasing in . The function satisfies where is the response function. The delay 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

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

#### Numerical Simulations

Case 1:

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 = 20 and = 15, respectively.

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

Case 2:

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

For the simulations below, = 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 = 10 and = 15.

MODEL 2

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

For this type of model, we consider the system

where > 0 is the rate of conversion of consumed prey to predator, > 0 is the death rate of the predator in the absence of the prey, is the constant rate of harvesting of the prey, and is the specific growth rate of the prey in the absence of the predators where and is continuous and decreasing in . The capture rate of prey per predator, that is the functional response is given by where . The delay is a nonnegative constant.

#### Example

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

with the equilibrium at = (40, 12).

#### Numerical Simulations

Case 1:

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.

Case 2:

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

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

MODEL 3

### Wangersky-Cunningham Model with Prey Harvesting

Consider the following system:

where denotes the growth rate of the prey population, is the death rate of the predator population, is the measure of effect of predation on , is the measure of effect of predation on , is the constant-rate harvesting of the prey.Also, where 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 , 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

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

#### Numerical Simulations

Case 1:

When the time delay , 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 = 2 and = 10.

Case 2:

The figures above are simulations for = 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 .

## 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 , 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 showing its dependence on . 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 , the faster the prey and predator populations go to the equilibrium . Incorporating a delay in the prey specific growth term gives a bifurcating periodic solution (e.g. = 0.826). As shown in the simulations, the prey and predator populations oscillates around the equilibrium. Furthermore, varying the harvesting constant changes the equilibrium value of predator population. The larger the , the smaller the , and the smaller the , the higher the value is.

In model 2, when , 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 = 7 and existence of a limit cycle for = 9, that is, a bifurcating periodic solution. As in model 1, the harvesting constant affects the dynamics of the system. Since, depends on , 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 , 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 = 0.05. Moreover, increasing decreases and decreasing increases . However, changing the values of the harvesting constant does not affect the general behavior of the solutions (in contrast to the effect of varying 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 and Additional Research

Many research studies and papers have cited Predator-prey models with delay and prey harvesting. 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.

Another paper that builds upon the work of Martin and Ruan is Bifurcation and control of a bioeconomic model of a prey–predator system with a time delay[5]. It was written by Kunal and Milan Chakraborty , with the purpose of taking the foundational model proposed in Predator-prey models with delay and prey harvesting and applying it to fisheries. It incorporated a gestationally-delayed predator, which made sense due to the transition of prey biomass to predator biomass. They compared two populations of prey, where one population underwent regular fishing and the other was protected.

The paper was heavily focused on maximizing yield for fisheries, as well as to propose new solutions to overfishing and overharvesting. Their goal in writing the paper was to reduce the lasting environmental impact of corporate fishing practices as well as mathematically assess the benefits of having no fishing zones in close proximity to free fishing areas. They found that in having those zones, fish were able to move in and out of fishing areas successfully.

## 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.

## 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
3. wikipedia site on [1]
4. Chakraborty, K., Chakraborty, M., & Kar, T. K. (2011). Bifurcation and control of a bioeconomic model of a prey–predator system with a time delay. Nonlinear Analysis: Hybrid Systems, 5(4), 613–625.

## See Also

For more on delayed predator-prey models, see the following: