Journal of Mathematics

Volume 2018, Article ID 2037093, 15 pages

https://doi.org/10.1155/2018/2037093

## A Reaction Diffusion Model to Describe the Toxin Effect on the Fish-Plankton Population

Department of Mathematics and Informatics, UFR/ST, Université Nazi Boni, 01 BP 1091, Bobo-Dioulasso 01, Burkina Faso

Correspondence should be addressed to Boureima Sangaré; rf.oohay@9791uozam

Received 10 December 2017; Accepted 5 March 2018; Published 10 May 2018

Academic Editor: Kenneth Karlsen

Copyright © 2018 Wendkouni Ouedraogo 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

This paper is aimed at the mathematical formulation, the analysis, and the numerical simulation of a prey-competitor-predator model by taking into account the toxin produced by the phytoplankton species. The mathematical study of the model leads us to have an idea on the existence of solution, the existence of equilibria, and the stability of the stationary equilibria. These results are obtained through the principle of comparison. Finally, the numerical simulations allowed us to establish a threshold of release of the toxin, above which we talk about the phytoplankton blooms.

#### 1. Introduction

Understanding the functioning of an ecosystem is a major issue for resource and environmental management. However, this goal remains difficult to attain due to the complexity of natural systems, especially in the aquatic environment, where many processes of all types interact with living organisms [1–3]. Plankton is the basis of all aquatic food chains and phytoplankton in particular occupies the first trophic level and the fluctuations in its abundance determine the production of marine environment. Rapid increase and almost equally rapid decrease separated by periods are the common features of plankton populations. In a broad sense, planktonic blooms can be divided into two types: “spring blooms” and “red tides.” Spring blooms occur seasonally for the changes in temperature or nutrient availability which are connected with seasonal changes in thermocline depth, strength, and consequent mixing. Red tides are localized outbreaks and occur due to high water temperature [4–7]. Toxic substances produced by phytoplankton species reduce the growth of zooplankton by decreasing grazing pressure and this is one of the important common phenomena in plankton ecology [2, 6, 8, 9].

Within the broad perspective drawn above, the present paper explores and compares the coupled dynamics of the phytoplankton and the zooplankton in a number of mathematical models. The system phytoplankton-zooplankton has attracted considerable attention from various fields of research [3, 8, 10, 11]. It is an important issue in mathematical ecology. The literature abounds in models focusing on various aspects of the problem. Recently, the attention has been focused on the role of the space in explaining heterogeneity and the distribution of the species and the influence of the spatial structure on their abundance [2, 6, 8]. However, the very question of the interactions between phytoplankton, zooplankton, and fish depending on space is far from being fully elucidated. As part of our work, we will highlight a threshold value of the toxin released by phytoplankton, below which the effect of the toxin influences less the dynamics of the zooplankton-fish system. To our knowledge, this has never been addressed in the literature. The proposed model consists of three interactive components, zooplankton, herbivorous fish, and toxin-producing phytoplankton, which reduce the growth of fish and zooplankton population. The model studied here is of the reaction-diffusion type, describing the dynamics of the phytoplankton-zooplankton-fish system in the sense of the work of Courchamp et al. [12].

The paper is organized as follows. As far as Section 3 is concerned, we will establish mathematical results such as the existence of a solution, stability of equilibria, and persistence, relating to the constructed model in the Section 2. Section 4 will be devoted to numerical experiments to illustrate the mathematical results. Finally, Section 5 is devoted to the conclusion and perspectives.

#### 2. Mathematical Model

In this section, we propose a model to describe the dynamics of the phytoplankton-zooplankton-fish system in the presence of toxin. We begin our modeling by a general model describing the dynamics of the prey-competitor-predator system, based on the equations with ordinary derivatives. And then we transform this model into a model of reaction-diffusion type while remaining in the logic of the works of Courchamp et al. [12] and Bendahmane [13]. It should be noted that the aim is to take into account the effect of the toxin on the fish population through that of zooplankton.

##### 2.1. Original Model Formulation

If we designate by the density of the prey, by the density of the competitor, and by that of the predator, according to [12], the general model is written as follows:where (i), , , , , , , and are positive functions,(ii) is the growth function of the prey,(iii) is the function that measures the mortality due to the competition between the same individuals of the prey,(iv) is the function that measures the mortality of the prey due to the competitor’s consumption,(v) is the competitor’s growth function,(vi) is the function that measures the mortality of the competitor from the release of the substance produced by the prey,(vii) is the function that measures the mortality of the competitor from the consumption of the predator,(viii) is the growth function of the predator,(ix) is the function that measures the external mortality of the predator population.

We continue our modeling by fixing the expressions of the functions intervening in model (1). The dynamics of the system are represented by Figure 1.