
MBW:Predatorprey models with delay and prey harvestingFrom MathBioThis is a simplified summary presentation of a paper entitled Predatorprey models with delay and prey harvesting by Annik Martin and Shigui Ruan published in the Journal of Mathematical Biology, Vol. 43, 247267 (2001) ^{[1]}. Contents
PredatorPrey Models with Delay and Prey HarvestingAbstractThree models on predatorprey 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 WangerskyCunningham 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. OverviewMathematics usedDelay Differential Equations and the corresponding solutions. Type of ModelPopulation Model Biological system studiedAny generalized system of the predatorprey relation. Further reading examples include Killer Whales and the Arthropod predatorprey model. BackgroundPredatorprey 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 predatorprey system is the arthropod system and can be found here. See also MBW:PodSpecific Demography of Killer Whales [1]. In practice, harvesting is based on maximum sustainable yield ^{[2]}. Overexploitation 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 Gausetype predatorprey model 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:
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. In system (3), the time delay can represent a gestation period or reaction time of the predators. System (4) depicts the wellknown 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 SimulationsGeneralized Gause Model with Prey Harvesting and Delay in the Prey Specific GrowthThe 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 constantrate 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. ExampleConsider the system where the equilibrium is at = (20,15). Numerical SimulationsThe 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. 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.
Generalized Gause Model with Prey Harvesting and Delay in the Predator Response FunctionFor 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. ExampleTo illustrate the dynamics of the model, let's consider the system with the equilibrium at = (40, 12). Numerical SimulationsIt 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. 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. WangerskyCunningham Model with Prey HarvestingConsider 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 constantrate harvesting of the prey.Also, where is a densitydependent 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. ExampleFor purposes of numerical simulations, let's consider the system which has an equilibrium at = (5, 68/5). Numerical SimulationsWhen 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. 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 DiscussionsIn 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 PredatorPrey 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/Predatorprey_model CitationsMany research studies and papers have cited the original of this paper. In particular, a research paper called On Nonlinear Dynamics of PredatorPrey 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 PredatorPrey 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 CodeThis 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 WangerskyCunningham Model with Harvesting are provided by example3.m. Further ReadingThe Dynamics of Arthropod PredatorPrey Systems References
