Abstract and Applied Analysis

Volume 2015 (2015), Article ID 241312, 7 pages

http://dx.doi.org/10.1155/2015/241312

## Birth Rate Effects on an Age-Structured Predator-Prey Model with Cannibalism in the Prey

^{1}CIMAT, Jalisco, s/n, 36240 Col Valenciana, GTO, Mexico^{2}Universidad Autónoma de Aguascalientes, Edificio 26, Avenida Universidad No. 940, 20100 Aguascalientes, AGS, Mexico

Received 9 September 2014; Revised 15 December 2014; Accepted 16 December 2014

Academic Editor: Wanbiao Ma

Copyright © 2015 Francisco J. Solis and Roberto A. Ku-Carrillo. 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 develop a family of predator-prey models with age structure and cannibalism in the prey population. It consists of systems of ordinary differential equations, where is a parameter associated with new proposed prey birth rates. We discuss how these new birth rates give the required flexibility to produce differential systems with well-behaved solutions. The main feature required in these models is the coexistence among the involved species, which translates mathematically into stable equilibria and periodic solutions. The search for such characteristics is based on heuristic predation functions that account for cannibalism in the prey.

#### 1. Introduction

The modeling of predator-prey populations is an intensive research area where a wide variety of models are proposed to describe and to analyze different mechanisms such as control of populations, oscillations of populations, delay maturation time effects, and pattern formation in predator-prey models with diffusion: see, for example, [1–10]. One relevant feature in these models is the predator-prey coexistence, which is of vital importance to obtain realistic models of predator-prey interactions. Mathematically, a first type of coexistence is established by stable equilibria which are the most basic solutions to the models. A second type of coexistence is given by stable periodic solutions and this is the most common accepted type in predator-prey ecological systems; however, it is technically more difficult to achieve.

Our first goal is to propose and analyze models derived from the classical approach suggested by Gurtin and MacCamy [5, 6] for predator-prey population models with age structure. Such approach has a starting point given by the McKendrick-Von Foester equation [11] and the generated models involve integrodifferential equations. By introducing appropriate birth rates, it is possible to obtain models given by systems of ordinary differential equations. An important characteristic of our proposed models is the consideration of the cannibalism effect on the prey, since several studies have reported the stabilizing consequences of cannibalism in predator-prey systems [12, 13]. There are numerous examples in nature where cannibalism is present, for example, the confused flour beetle (*Tribolium confusum*) [14], the red flour beetle (*Tribolium castaneum*) [15], and the Dungeness crab (*Metacarcinus magister*) [16].

Our second goal is to analyze the relationship between the existence of periodic solutions and the functional form of the birth rate. We require analytic solutions of the models or that their potential singularities consist only in removable poles. Such process, known as* analytical modeling*, was introduced in [17]. In such work, the required conditions leading to special nonlinearities that discard traditional interaction terms like the law of mass action and logistic terms were established. Examples were exhibited with predation functions where coexistence between the prey population and the predator population was achieved. One possible drawback associated with this new approach is that in order to achieve coexistence, restrictive conditions have to be imposed on the systems. Hopefully, simpler models can be thoroughly analyzed and then built upon to derive increasingly more complex models and this is the basic approach that we will use here.

Before attempting to model the vital-rate changes, we need to address the critical lack of age-structured information. To do this, we propose in Section 2 a parametrized birth rate which generalizes the one presented in previous works. Then in Section 3, we use a general framework in order to derive predator-prey models with age structure based on the new birth rate. In Section 4, we analyze the models including numerical simulations and interpret the biological implications of the results. Finally, conclusions are given in Section 5.

#### 2. Birth Rate Prey Analysis

Population models with age structure are commonly used to analyze the evolution of the dynamics of animal populations with delayed maturation and long life spans, among others. One of the primary building blocks for such population models is the age-specific birth rate. The birth rate is usually the dominant factor in determining the rate of population growth. It depends on the level of fertility, the age structure of the population, and many other factors. The qualitative properties of a birth rate include the fact that it is expected to be small for newborns and old individuals, whereas for young adults it must have the shape of unimodal function with a maximum value for matured adults. In our case we assume that the birth rate depends only on the age of population. Mathematically, a birth rate denoted by is a continuous function defined only for nonnegative ages; that is, Furthermore, it is a positive bounded function with compact support. Since the birth rate for old individuals is 0. That is, , for and for all for some constants and

There are several functional forms of the function that have been introduced in the literature. For example, in [18], we have introduced an appropriate reproductive rate of the prey population given by with and This function does not have compact support, but for large values of , it is practically zero. Let us recall that if the birth function (1) behaves in a way that is appropriate for many mammals and if , then has been used for certain fish species. In this case vanishes for equal to zero and it approaches zero for large values of . It reaches a global maximum at the age equal to . Here, we propose a parametrized birth rate which includes (1) as a particular case. This new rate is given bywhere is a natural number. Notice that the new rate has mathematically and biologically all the required properties, since it is practically zero for larger values of . See Figure 1, where we plot the graph of for some specific values of . It is important to clarify that the new birth rate, with , an integer, will allow us to obtain a differential equation system instead of an integrodifferential system as we will discuss in the next section.