International Journal of Differential Equations

Volume 2018, Article ID 2381483, 13 pages

https://doi.org/10.1155/2018/2381483

## The Impact of Price on the Profits of Fishermen Exploiting Tritrophic Prey-Predator Fish Populations

Analysis, Modeling and Simulation Laboratory, Hassan II University, Casablanca, Morocco

Correspondence should be addressed to Youssef El Foutayeni; moc.liamg@ineyatuof

Received 21 August 2017; Accepted 26 November 2017; Published 1 January 2018

Academic Editor: Abid A. Lashari

Copyright © 2018 Meriem Bentounsi 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 define and study a tritrophic bioeconomic model of Lotka-Volterra with a prey, middle predator, and top predator populations. These fish populations are exploited by two fishermen. We study the existence and the stability of the equilibrium points by using eigenvalues analysis and Routh-Hurwitz criterion. We determine the equilibrium point that maximizes the profit of each fisherman by solving the Nash equilibrium problem. Finally, following some numerical simulations, we observe that if the price varies, then the profit behavior of each fisherman will be changed; also, we conclude that the price change mechanism improves the fishing effort of the fishermen.

#### 1. Introduction

The problem of modelization is, perhaps, the most challenging in modern ecology, biology, chemistry, and many other sciences. In population dynamics, specially, in the dynamics of prey-predator marine species interactions modelization has gained a great importance. A predator is an organism that feeds another organism. The prey is the organism which the predator feeds. Predator always depends upon its prey and the predator dies if it does not get food. The first basic classic prey-predator model is renowned by Lotka-Volterra model and mathematical formulation of this model is directly related to the great work of Lotka (in 1925) and Volterra (in 1926). Thanks to this prey-predator model, other models have been proposed and studied [1–3]. In [1], the authors have considered predator-prey dynamics with predator “searching” and “handling” modes; they have derived a model that generalizes Holling’s functional responses and they have proved results concerning local and global properties, including for oscillations. In [2], the authors have formulated and studied a stage-structured predator-prey model of Beddington-DeAngelis type functional response to investigate the impact of predation over the immature prey by the juvenile predator. In [3], the authors have studied the global stability of diffusive predator-prey system of Holling-Tanner type in a bounded domain.

Let us add that many researchers have studied extended tritrophic (prey, middle predator, and top predator) models to understand the interaction of different types of species [4–6]. In [4], the dynamics of a predator-prey model with disease in super predator are investigated. In [5], the authors have studied a prey-predator model with the concept of super predator under economic perspective. In [6], the authors have made a systematic analysis of the dynamics of a predator-prey system with type II functional response, in which the predator growth rate is affected by the presence of a super predator.

In recent years, the biodiversity of marine populations is threatened by human impact, more precisely, by harvesting, which required many scientists to study bioeconomic models of fishery [7, 8]. In [7], the authors have made a mathematical study of a bioeconomic model of fishing for multisite, exploited by several fishermen, except one of them which is defined as not exploitable free fishing zone. In [8], the authors proposed and analyzed an extended model for the prey-predator-scavenger in presence of harvesting to study the effects of harvesting of predator as well as scavenger.

In this paper, we have studied a tritrophic (prey, middle predator, and top predator) generalist model. The objective is to calculate the fishing effort that maximizes the profit of the fishermen, while respecting the conservation of the three fish populations, and also to study the effect of the variation of the price on each profit. The remaining part of this paper is organized as follows. In Section 2, we introduce the biological tritrophic model. The existence and the stability of the steady states solutions are analyzed in Section 3. The bioeconomic model of the prey, middle predator, and top predator system is proposed in Section 4. In Section 5, we compute and solve the Nash equilibrium problem. In Section 6, we solve the linear complementarity problem. In Section 7, we present some numerical simulations to show the impact of price on the profits of fishermen. Finally, a brief conclusion is given in Section 8.

#### 2. Biological Model

In this section, we consider a tritrophic prey-predator model which consists of three constituent populations, that is, prey, middle predator, and top predator. We impose that the population of prey grows in the logistic manner with birth rate constant and there exist interactions between the prey and middle and top predator due to defensive ability of prey; we impose that the population of middle predator grows also in the logistic manner with birth rate constant, prey is favorite food for middle predator , and hence in the presence of favorite food the population density of middle predator will increase, and there are interactions between the middle predator and top predator due to defensive ability of middle predator; in the presence of favorite food (prey and middle predator) of top predator the population density of top predator will increase. Hence we can write this model in mathematical terms aswith positive initial conditions , , .

Here , and are the per capita growth rate of prey, middle predator, and top predator, respectively; , , and are the maximum value which per capita reduction rate of and can attain, respectively; is the conversion rate of prey into middle predator , and and are the conversion rate of prey into top predator and the conversion rate of middle predator into top predator , respectively.

#### 3. The Steady States of the System

Since the focus is on the growth of marine species, there is need for the steady states of the system to satisfy conditions for nonnegativity. Furthermore, it is realized that the predators cannot survive in the complete absence of their prey. System (1) has eight biologically feasible nonnegative steady states. These steady states are obtained by solving the system of equations(i), , , .(ii), where and .(iii), where and .(iv), where and .(v), where where .

One can see that the steady state equilibrium exists if , , and

##### 3.1. Analysis of Steady States

The Jacobian matrix for system (1) is given by where

At any steady state solution, the Jacobian matrix is computed. Let denote the Jacobian evaluated at for , the corresponding entries, and denote the Jacobian evaluated at .

###### 3.1.1. Local Stability of the Steady State

For the equilibrium point the Jacobian matrix is given by The eigenvalues are found to be , , and , and then this point is unstable.

According to Table 1, Figure 1 shows the dynamical behaviors and phase space trajectory of the prey, middle predator, and top predator fish populations against time, beginning with the initial values , , and . By Figure 1 we find that the steady state point is unstable, and more precisely this point tends to the point .