Abstract
Although many kinds of numerical methods have been announced for the predator-prey system, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, in this paper, a new interpolation collocation method is proposed for a class of predator-prey systems with complex dynamics characters. Some complex dynamics characters and pattern formations are shown by using this new approach, and the results have a good agreement with theoretical results. Simulation results show the effectiveness of the method.
1. Introduction
Mathematical models have played a significant role in understanding the dynamics of populations for over a century. These models have helped abstract key features of interactions between biological organisms that have given insight into a variety of observed phenomena in real populations.
In [1], let and be the population size of a prey and predator species, respectively, and suppose that these functions obey the Gause-type model:
In [2], the authors give a predator-prey population dynamics. This model includes all three components (predator, prey, and subsidy) in a single spatial location:
Turing [3] proposed a dynamical mechanism, which has been extensively used to explain how Turing patterns are formed, and it is now known as Turing bifurcation or Turing instability. Reaction-diffusion systems have attracted increasing attention from the mathematical biologists in recent years to seek insights into the fascinating patterns that occur in living organisms and in ecological systems. As suggested in [2], a more realistic system could account for continuous spatial variation in the populations. In[4, 5], the authors assumed that these populations disperse randomly over a spatial region. This leads to the following set of predator-prey systems with diffusion:where and are the prey and predator populations, is the quantity of subsidy, t is time, are all nonnegative kinetic parameters, and , , and are the positive diffusion coefficients for the three populations; .
The sufficient condition of locally asymptotically stability of the equilibrium has been obtained in [4–6]. Turing bifurcation occurs when the equilibrium state that is stable in absence of nondiffusion becomes unstable in presence of cross-diffusion.
In the present paper, we consider the following reaction-diffusion system [7–9], which is described aswhere , and are the unknown functions. , , the smooth boundary is , and homogeneous Neumann boundary condition, namely, . The parameters , , and are the diffusion coefficients, and is the Laplacian operator, respectively. Parameter is the known parameter. are the known nonlinear functions of .
In recent years, reaction-diffusion systems [10, 11] have attracted increasing attention from the mathematical biologists to seek insights into the fascinating patterns that occur in living organisms and in ecological systems [5, 12, 13].
In general, the exact solution of system (4) cannot be obtained, and the approximate solution must be obtained by numerical calculation. Although many kinds of numerical methods of the nonlinear reaction-diffusion system have been announced, such as finite difference method [14], B-spline method [15], finite element method [16, 17], spectral method [18–20], the perturbation method and variational iteration method [21, 22], barycentric interpolation collocation method [23–26], and reproducing kernel method [27, 28], this paper investigates some nonlinear diffusion predator-prey systems [5, 12, 13] based on a new interpolation collocation method, and the model (21) is adopted as an example to elucidate the solution process.
2. Bifurcation Analysis of the System
In this section, we give the Turing bifurcation conditions of system (4). We assume an equilibrium point of nondiffusive system (4) as , so
Now, we do linear stability analysis of the equilibrium point . We evaluate the Jacobian matrix of the system at as
In general, if real part of eigenvalues of is negative, then nondiffusive system (4) is stable. Next, we derive diffusion-driven instability conditions and show that these are a generalization of the classical diffusion-driven instability conditions in the absence of diffusion. The Jacobian matrix of system (4) is
Turing bifurcation occurs when the equilibrium state that is stable in absence of nondiffusion becomes unstable in presence of cross-diffusion. Thus, the only way for the to become an unstable point of cross-diffusion system (4) is that the real part of eigenvalues is positive; then, diffusion system (4) is unstable.
3. Description of the Method
3.1. Periodic Sinc Function
Sinc functions are used in different areas of physics and mathematics. A periodic sinc function is defined aswhere and is the interpolation function of periodic δ function.
It can be proved that , and the first-order derivative of function at is
The two-order derivative of function at is
Given , we define the following interpolation space:
Let be the interpolation operator for any function defined on . The interpolation function of sequence can be written aswhere . It is not difficult to derive simple expressions of the nth derivatives of at :where . can be written in the matrix form as follows:
Noting the nth differential matrix :where the 1st differential matrix is as follows:and the 2nd differential matrix is
3.2. Interpolation Collocation Method
To solve system (4), we consider a regular region , and the interval is divided into N different nodes. , . Using periodic sinc function (12), we first approximate , , and as
At collocation nodes , the following relations hold:
Notingwhere ,
Therefore, formula (19) can be written in the matrix form as follows:where is the Kronecker product of matrix and and , is the N order unit matrix, respectively.
Employing equations (18)–(21), the discrete form of equation (4) can be written as
Here,
Thus, we have the following system of ODEs:in which
The RK-4 for ODEs (23) is given aswhere . An initial condition is required for starting the integration process and at each time level, boundary conditions should be applied as follows:in which, denotes the boundary operator.
3.3. Stability of the Method
The stability of system (23) can be discussed according to the eigenvalues of spatial operator . Let the eigenvalues of be ; then, one of the following conditions should be fulfilled for the stability of system (23):(i)If all are real and (ii)If all are pure imaginary and (iii)If are complex, then should be in the stability boundary of the RK-4 method
It is stated that in [29], there may be some tolerance that real parts of eigenvalues may be small positive numbers if the eigenvalues are complex.
4. Numerical Experiment
In this section, some numerical experiments are studied to demonstrate the accuracy of the present method.
Experiment 1. We consider system (3). For, bifurcation analysis of Experiment 1, see references [4–6]. Using the present method, taking , the parameter , , , , , , , , , , , , and . Numerical solution and pattern of Experiment 1 with the different initial conditions and initial condition at the different t are shown in Figures 1–7.
The present method could be extended for use in solving other nonlinear cross-diffusion systems.
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
Experiment 2. Let us consider the following nonlinear diffusion system:
4.1. Bifurcation Analysis of Experiment 2
In this section, we derive non-cross-diffusion-driven stability conditions.
If , then the nondiffusive Experiment 2 has three constant equilibria: the zero equilibrium ; the boundary equilibrium ; and the positive equilibrium with .
Now, we do stability analysis of the equilibrium point . The linearized system of Experiment 2 at iswith Jacobian matrix iswhere is often called the wave number, and we have the following eigenvalue of :where
From equations (31)–(33), it is clear that if holds, then It is easy to see that if and only if the condition holds. Therefore, without diffusion, i.e., whenhold; the positive equilibrium of Experiment 2 is locally asymptotically stable.
In this section, we derive cross-diffusion-driven instability conditions and show that these are a generalization of the classical diffusion-driven instability conditions in the absence of cross-diffusion.
We assume that This implies that the product of the self-diffusion coefficients and should diffuse much faster than the product of the cross-diffusion coefficients and .
From equations (31)–(33), obviously, if , holds. In this case, the positive equilibrium is stable and the Turing bifurcation cannot occur. Thus, the Turing bifurcation may occur only when . Choose as the bifurcation parameter.
From equations (31)–(33), Turing bifurcation at , and the parameter is determined by the following equality:
Based on the above analysis, we obtain if and the condition holds, then the positive equilibrium of Experiment 2 is locally asymptotically stable for and unstable for .
Based on the above analysis, the sufficient condition of locally asymptotically stable of the equilibrium is obtained.
Theorem 1. Suppose that and the condition holds.(i)If and , then the positive equilibrium of Experiment 2 is locally asymptotically stable(ii)If and , then the positive equilibrium of Experiment 2 is locally asymptotically stable for any and (iii)If and , then the positive equilibrium of Experiment 2 is locally asymptotically stable for and unstable for
Bifurcation diagram of Experiment 2 with the different parameters are shown in Figures 8 and 9. In Figures 8 and 9, hopf curve 1 is Turing curve 1 is , and hopf curve 2 is .
4.2. Numerical Simulation of Experiment 2
Using the present method, taking , the parameter , , , , , , and . Numerical solution and pattern of Experiment 2 with the different initial conditions at the different t are shown in Figures 10–16.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
5. Conclusions and Remarks
In this paper, a sinc function interpolation collocation method is proposed for a class of nonlinear diffusion predator-prey systems. Some complex dynamics characters and pattern formations of predator-prey systems with the same parameters and different initial conditions are shown by using this new approach. Simulation results show the effectiveness of the new method. The present method could be extended for use in solving other nonlinear reaction-diffusion systems. In further work, we will be devoted to studying some fractional-order predator-prey systems.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This paper was supported by the Natural Science Foundation of Inner Mongolia (nos. 2017MS0103 and 2019BS01011), the Science Research Project of Higher Education in Inner Mongolia Autonomous Region (no. NJZY20158), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (no. NJYT-20-B18), and the National Natural Science Foundation of China (no. 11361037).