The protection zone of biological population

Abstract

Many species are endangered, if we do not act fast, we are going to lose them forever. Establishing protection zone is widely used to protect endangered species. How is the effect of this method? What factors affect the effect of the protection zone? We study this topic by a new mathematical model. We examine the effect of the protection zone and conclude that the protection zone is effective for conservation of population resources and ecological environment, though in some cases the extinction cannot be eliminated. The dangerous region, the parameters domains and the typical bifurcation curves of stability of steady states for the considered system are determined. Our results provide theoretical evidence for the practical management of biological resources.

Introduction

Biological resources are renewable resources. In the exploitation of population resources, both the economic benefits and environmental effects should be taken into account. However, with the development of society and economy, environmental pollution and ecological disruption have become serious social problems. Many biological resources have been exploited unreasonably. Because of the influence of the nature factors and the artificial factors, district ecosystem environment is continuously deteriorating. Many species of animal had die out, several other species are endangered. The consequence is unimaginable if we do not take effective measures immediately. So, in order to prevent the biological resources from destruction and preserve the ecological environment, various measures have been proposed. The method of establishing protection zone for biological resources is applied popularly. Like many other problems, an appropriate mathematical model can make a deeper understanding of the internal mechanism as well as the law of development. An appropriate mathematical model can also provide us a better understanding of the effect of different parameters in a system. Some scholars have used mathematical models to research the effect of the protection zone [1], [2], [3], [4]. In our paper, we will use a new mathematical model to discuss the effect of the protection zone which is different from other models.

When we consider the effect of the protection zone, the region $\Omega$, where the species live in, is divided into two subregions ${\Omega }_1$ and ${\Omega }_2$; ${\Omega }_1$ is the natural environment and ${\Omega }_2$ is a protection zone in which there is plenty of food, adequate medical care, and hunting and capture of biological resources are prohibited. Therefore, the difference of population density between ${\Omega }_1$ and ${\Omega }_2$ exists; thus, the diffusion can occur between ${\Omega }_1$ and ${\Omega }_2$, which is assumed to be proportional to the density difference and the proportional coefficient is assumed to be $d(>0)$. The density of population in ${\Omega }_1$ and ${\Omega }_2$ is denoted by $x\left(t\right)$ and $y\left(t\right)$, respectively. In the literature [1], authors gave a model on harvested population with diffused migration:

$$\{\begin{array}{c}\frac{dx}{dt}=rx(1-\frac{x}{K})-d(x-y)-Ex\hfill \\ \frac{dy}{dt}=ry(1-\frac{y}{K})+d(x-y)\text{.}\hfill \end{array}$$

In this model, we cannot see the effect of the size of the protection zone. However, it is well known that the size of the protection zone do affect the protective results. Then we want to know what is wrong with this model; the answer is the diffusion term. According to the laws of physics and biology, the diffusing capacity is proportional to the difference in concentrations. So, $d(x-y)$ represents the diffusing capacity, that is the total biomass caused by the diffusion effect. However, $x$ and $y$ represent the densities of population in ${\Omega }_1$ and ${\Omega }_2$ (they must be the densities of population or we cannot use the law of diffusion). Therefore, we cannot add them directly since they have different units. In order to solve this problem, we consider the size of ${\Omega }_1$ and ${\Omega }_2$. Assume that the size of ${\Omega }_1$ is $H$ and the size of ${\Omega }_2$ is $h$. (It seems that many scholars ignored this problem; in many documents they added them directly for different models in many fields [1], [5], [6], but the results are still right since we can regard it as an exceptional case. In the literature [7], the authors have taken note of this problem and given a model about marine reserves in which the variables are in terms of the stock of the populations. In our paper, we will give a model of the protection zone in terms of the density of populations, and the same idea can be used to improve many useful models.) Then, system (1) can be improved as

$$\{\begin{array}{c}\frac{dx}{dt}=rx(1-\frac{x}{K})-\frac{d}{H}(x-y)-Ex\hfill \\ \frac{dy}{dt}=ry(1-\frac{y}{K})+\frac{d}{h}(x-y)\text{,}\hfill \end{array}$$

where $K$ is the carry capacity of the environment, $r$ is the intrinsic growth rate, in our model, and $E$ is the comprehensive effects of the unfavorable factors of biological growth relative to the biological growth in the protection zone, such as the capturing by predator, lacking of the food, medical assistance and so on.

In the next section, we will first analyze the effect of the size of the protection zone when the size of the protection zone is extremely large or small. Then we will study the population’s dynamical behavior and analyze the existence and stability of the equilibrium points. The dangerous region, the parameters domain and the typical bifurcation curves of stability of steady states for the considered system will be determined. Next, we will investigate the practical effects of the protection zone on the conservation of population resources and indicate the biological significance of the main results.