Journal of Applied Mathematics

Volume 2018, Article ID 6146027, 10 pages

https://doi.org/10.1155/2018/6146027

## The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator

Department of Probability Theory and Mathematical Statistics, Riga Technical University, Kaļķu iela 1, Riga LV-1658, Latvia

Correspondence should be addressed to Kārlis Šadurskis; moc.liamg@siksrudas.silrak

Received 8 October 2017; Accepted 23 January 2018; Published 20 June 2018

Academic Editor: Said R. Grace

Copyright © 2018 Jevgeņijs Carkovs et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quantitative analysis of the above model based on the probabilistic limit theorem. If the predators’ gains and the time intervals between predator attacks are sufficiently small, our proposed method allows us to derive an approximative average dynamical system for mathematical expectations of population dynamics and the stochastic Ito differential equation for the random deviations from the average motion. Assuming that the averaged dynamical system is the classic Holling type II population model with asymptotically stable limit cycle, we prove that the dynamics of stochastic model may be approximated with a two-dimensional Gaussian Markov process with unboundedly increasing variances.

#### 1. Introduction

One of the most popular models of the dynamics of interacting predator and prey populations, such as those found within invertebrate and similar domains of life, in mathematical biology is the system of ordinary differential equations proposed in [1]:where phase variables and denote the density of prey and predator populations, respectively. In this model it is assumed that in the absence of a predator the prey population has a potential carrying capacity and develops according to the logistic law with an intrinsic growth rate , and in the absence of a prey the predator population exponentially decreases to zero with the intrinsic growth rate . The mutual influence of changes in the densities of the prey and predator in model (1) is considered by the trophic function , where the positive parameter is the prey consumption rate by the predator or, in other words, corresponds to the number of prey individuals that can be “eaten” per unit of time. The positive parameter reflects the saturation of the amount of prey consumed and, in addition, depends on the rate of reaction of the predator, i.e., the time between attacks on the prey. The parameter in formula (1) is a conversion factor that determines the effect of “eaten prey” on the growth rate of the population of the predator. The popularity of the model (1) is explained by the fact that under certain assumptions about positiveness of parameters , and it is structurally stable [2]; that is, there exists a unique asymptotically stable periodic trajectory. This model describes stable fluctuations of the size of the predator and prey populations that are sometimes observed in biological ecosystems. In accordance with the continuous type dynamic system (1), both populations are in permanent contact, and the benefits gained and losses suffered by predator and prey correspondingly in an arbitrary small time interval are proportional to the length of the interval . In fact, it is clear that changes in the size of both populations are accidental and can only be modelled on average by formula (1). Therefore, subsequently papers were published that took into account the random nature of the model under study by adding stochastic terms of the white noise type in the system of (1) (see the review in [3]). It is important to note that the choice of this type of random perturbation allows preserving the principle of predictability of the behaviour of populations, since stochastic differential equations define random Markov type processes. In the case of modelling by the means of stochastic differential equations, the predator’s gain and the prey’s loss during time contain not only terms proportional to but also terms that are proportional to the increment of the Brownian motion process . In most of these papers, the authors manage to prove the possibility of the existence of a stable stationary close to periodic ergodic process that describes the behaviour of the stochastic model under sufficiently small random perturbations.

In the modern literature on mathematical biology many authors use stochastic differential equations as a mathematical model for predator-prey ecological systems (we refer again to the review in [3]). The most typical papers using stochastic models of predator-prey populations are [4–16]. The authors of these papers study the effect of stochastic perturbations of various parameters of classical models, adding either white noise to a chosen parameter [4–10] or the integral over a centered Poisson measure [11–16]. Using the apparatus of modern stochastic analysis, authors study the possibility of the existence of positive solutions, the stability of possible stationary solutions, and the existence of moments of solutions, as well as estimating the asymptotics of the moments of solutions and other properties of solutions. It should be noted that most of the aforementioned papers deal with models with stochastic additives containing no higher than second-order phase variables. If, however, the diffusion coefficients or the integrand of the integral over the Poisson measure has a more complicated form, then the apparatus proposed in the aforementioned papers can hardly be used. At the same time, when analysing the dynamics of some predator-prey biological communities, it may be necessary to investigate the consequences of nonpermanent random contacts that occur at random time moments. In this case, it is natural to assume that at the time of the contact extraction of prey by predator will be a random process that is proportional to the functional response and depends on the phase coordinates in a more complex form.

However, in this kind of model, the predator’s gain and the prey’s loss during time are proportional to the normally distributed random variable with parameter and therefore can be either positive or negative, which is poorly consistent with the definition of these terms in a formula of type (1). In this paper we propose a model that also makes it possible to take into account the stochastic character of the trophic function of the dynamics of populations as considered in the Holling type II model, but the predator’s gain and the prey’s loss are limited positive random variables. Besides, our model takes into account the possibility that the predator may take some time to attack the prey, and therefore there are intervals when both populations develop independently. Moreover, our model also fulfils the Markov property.

We propose a method of approximate numerical analysis of the probabilistic characteristics of population dynamics, based on application of the asymptotic diffusion approximation algorithm [17]. Application of the diffusion approximation method for the asymptotic estimation of the probability characteristics of the Markovian dynamical system, analogously to the algorithm of the classical central limit theorem, consists of two steps. Initially, using the small parameter* epsilon* and the limit theorem for sequences of Markov processes, we find a deterministic dynamical system for the averaged phase variables. This is followed by finding a stochastic dynamical system for normalized deviations from solutions of the averaged dynamical system. The resulting stochastic differential equation of the Ornstein-Uhlenbeck type is well studied and may be relatively simply analysed [17].

#### 2. The Model

Let us propose a model where the contacts between predator and prey occur very often at random time moments , which are the moments of discontinuity of the trajectories of a stationary piecewise constant Poisson process [18] given on a certain probability space with an exponentially distributed length of the constancy intervals , where , and is a small positive parameter. Let us denote as the minimal filtration [19] to which the process is adapted. Let us also assume that at time moments the random variables have a continuous distribution and, therefore, without any loss of generality, we may consider that this distribution is uniform on the interval . It is known [18] that the probability distributions of a homogeneous Markov process are uniquely determined by its weak infinitesimal operator given byIn our case the Markov process is given by an infinitesimal operatorwhere is a positive parameter.

Let us proceed to the formal definition of the Markov dynamical system dealt with in this paper. Let us suppose that the time of observation starts at , and densities of prey and predator populations at this moment are , respectively. Suppose that the size of the prey’s population changes in accordance with the logistic equationAnd the size of the predator’s population changes by the rulewhere , , , and . At the time moment the collision between predator and prey individuals occurs, and predator’s gain is given by the expression Here and further . Up to the time moment , the dynamics of the size of the prey and predator populations are also given by (5)-(6) with initial conditions and . At the moment the predator finds the prey again and its gain is given by expression (6) where the argument is replaced by . Then up to the moment the sizes of predator and prey populations change again according to law (4)-(5), etc.

It is useful to note that, using terminology and results of monograph [20], Markov dynamical system (3)-(4)-(5)-(6) may be defined by the system of stochastic differential equationswhere is a Poisson random measure with the parameter .

Figure 1 shows the trajectory of the solution of the system of the impulse-differential equations (4)-(5)-(6) with the initial conditions and and the parameter values , , , , , , and . Figure 2 shows the trajectory of the solutions of the system of (1) with initial conditions and and with the same the parameter values , , , and .