Feedback stabilization for prey predator general model with diffusion via multiplicative controls

1.   Introduction

The dynamics between prey and predator species in natural systems can be represented by a coupled system of nonlinear reaction-diffusion equations. This type of system is usually represented by a system of partial differential equations, rather than a system of ordinary differential equations, which are often used to represent interactions between prey and predator populations without diffusion (see for instance, ^[13] and ^[30] and references therein). We are concerned with the following mathematical model, which expresses the conservation of predator and prey densities:

where $y_1$ and $y_2$ represent the prey and predator population densities at time $t$ respectively. The function $y_1 h\left(y_{1}\right)$ is the intrinsic growth rate of the prey $y_1$ and signifies its growth rate in the absence of the predator. It can be linear if $h\left(y_{1}\right) = r_{1}$, logistic if $h\left(y_{1}\right) = r_{1}\left(1-y_{1} / \kappa_{1}\right)$, Gompertz if $h\left(y_{1}\right) = h_{0} \ln \left(\kappa_{1} / y_{1}\right)\left(h_{0 }, r_{1}, \kappa_{1} > 0\right.)$ etc. See ^[1,2] and the references therein. The predator's functional response to prey is $y_1k(y_1)$, which represents the number of prey individuals consumed per unit area and unit time per predator. It includes as particular cases of various classical functional responses: $y_{1} k\left(y_{1}\right) = \beta y_{1}$ (Holling type Ⅰ), $y_{1} k\left(y_{1}\right) = \beta y_{1} /\left(1+\mu y_{1}\right.$) (Holling type Ⅱ), $y_{1} k\left(y_{1}\right) = \beta y_{1}^{2} /\left(1+\mu y_{1}^{2}\right.$) (Holling type Ⅲ; see ^[3]), $y_{1} k\left(y_{1}\right) = \beta y_{1} /\left(\gamma+\mu y_{1}+y_{1}^{2}\right)$ (Holling type Ⅳ; see ^[4]) etc. Here $\beta, \gamma > 0, \mu \geq 0$.

There is a large amount of literature related to the mathematical study of prey-predator systems of the diffusion type. In ^[24], Morita and Tachibana showed the existence of an entire solution (i.e., a solution that exists for all $(t, x)\in\mathbb{R}^2)$ of the predator-prey reaction-diffusion systems with Holling type Ⅰ. The proof is carried out by applying the comparison principle and an appropriate pair of a subsolution and a supersolution. For Holling Ⅱ functional response, Garvie and Trenchea proved the existence of a solution by using semigroup theory and application of the invariant region method of Smoller (see ^[15]). In ^[1], Apreutesei and Dimitriu studied the well-posedness of a predator-prey reaction-diffusion system with Holling type Ⅲ. They show the existence of the solutions provided that the initial data are positive and satisfy a specific regularity. For more results on the well-posedness of these systems, we refer, for instance, to ^[8,9,10] and the reference therein. Recently, Mi et al. ^[23] considered a nonlocal predator–prey model with double mutation; they defined pair of upper and lower solutions, and they designed a new comparison principle that ensures the existence of the solutions.

In our case, the situation is quite complicated due to the generalized nonlinearity considered, which encompasses all Holling functional responses nonlinearities. Consequently, we use the Banach fixed point and the stabilization of an associated system to obtain the global well-posedness. The existence and stability results concerning the steady states of these systems have been extensively studied see, for example, ^[2,5,7,21] and the references therein. The results show that the system exhibits very unusual behavior for some parameter values, while some steady states are stable under system parameter constraints. It would be interesting to investigate a method that allows driving the systems to these equilibrium states without adding additional constraints on the system parameters that might contradict the measures taken during the modelling. To this end we use multiplicative controls to stabilize the system, this choice is determined by the real application. In fact, for the prey predator model, these controls can be interpreted as harvesting efforts. A huge amount of literature has been devoted to stabilizing uncoupled linear systems via multiplicative controls. For example, we mention the works of ^[4,17,26] and the references therein. However, the choice of such control in the nonlinear cases generates new difficulties. Indeed, the first difficulty is that the stabilization of nonlinear parabolic coupled systems by using multiplicative controls remains an open problem. Another difficulty of the problem lies in the fact that these controls are nonlinear, which doubles the nonlinearity of the system. Note also that the controllability of bilinear systems is an open problem see ^[3]. For hyperbolic coupled systems we refer to ^[19], where the authors characterize the stabilization of a class of coupled hyperbolic systems by using multiplicative controls. They showed the equivalence between stabilization and the observability of the uncontrolled system. In this work, we hope to achieve the stabilization result. More precisely, let

We obtain from (1.1) the following system

We say that $(y^{e}_1, y^{e}_2)$ is an equilibrium state of (1.3) if and only if:

1- $(y^{e}_1, y^{e}_2)\in H^2(\Omega)\cap H^1_0(\Omega)$.

2- $(y^{e}_1, y^{e}_2)$ solves the following elliptic system

Translate $(y^{e}_1, y^{e}_2)$ into zero via the following change of variable $z_1 = y_1-y_1^{e}$ et $z_2 = y_2-y_2^{e}$. Obviously $(z_1, z_2)$ solves the following system

then stabilizing (1.3) towards $(y^{e}_1, y^{e}_1)$ is reduced to the stability of the null solution to system (1.4). By injecting a multiplicative control into the prey and predator equations, we obtain from (1.4) the following system

where $v(t)$ is a feedback control to be determined. The control operator $B$ is assumed to be bounded from $L^{2}(\Omega)$ to $L^{2}(\Omega)$. The terms $v(t)Bz_1$ and $v(t)Bz_2(t)$ can be regarded as the effort applied to harvest the prey and predator, respectively. System (1.5) represents the evolution of predator-prey densities under the action of harvesting. In the following, we construct the the control feedback $v(t)$, ensuring the exponential stabilization of (1.5).

The rest of the paper is organized as follows. In section 2, we present the assumptions and main results. We start with the stabilization of the linearized system and extend the result to the nonlinear system, where we show the well-posedness and exponential stabilization using Banach fixed points. In section 3, we illustrate the obtained results for different Holling responses.

2.   Assumptions and main results

For a bounded open set $\Omega \subset \mathbb{R}^{N}$, we denote by $H$ the Lebesgue space $L^2(\Omega)$ endowed with the inner product $\langle., .\rangle$ and its corresponding norm $\|.\|$, $\mathcal{H}$ the Cartesian product $L^2(\Omega)\times L^2(\Omega)$ with the norm $\|.\|_{\mathcal{H}}$ and $\mathbb{H} = H^{1}(\Omega)\times H^{1}(\Omega)$ with the norm $\|.\|_{\mathbb{H}}$.

The following assumptions will be in effect everywhere in the following:

$(H_1)$ $(y^{e}_{1}, y^{e}_{2})\in C(\overline{\Omega})$

$(H_2)$ $f, g \in C^{1}(\mathbb{R} \times \mathbb{R})$ satisfy the growth condition

where $m_0$ is a positive integer and $r_i, 1 \leqslant i \leqslant m_0$, are such that

$(H_3)$ $y_1(t, x)\geq 0$ and $y_2(t, x)\geq 0$ provided that $y^{0}_{1}(x)\geq 0$ and $y^{0}_{2}(x)\geq 0$.

Assumptions $(H_1)$ and $(H_2)$ for Holling types Ⅰ, Ⅱ, Ⅲ and Ⅳ, imply, in particular that: $f_y(y^{e}_{1}, y^{e}_{2}), f_z(y^{e}_{1}, y^{e}_{2}), g_y(y^{e}_{1}, y^{e}_{2}), g_z(y^{e}_{1}, y^{e}_{2})\in L^{\infty}(\Omega)$. Assumption $(H_3)$ is proved for Holling type Ⅰ, Ⅱ, Ⅲ and Ⅳ; see, for instance, ^[1,8].

Consider System (1.5),

The linearized system associated with (2.2) is given by

2.1.   Exponential stabilization of the linearized system

In this section, we establish the exponential stabilization of the linearized system (2.3).

Spectral proprieties

Let $a = \sup_{x\in \Omega}|f_y(y_1^{e}(x), y_2^{e}(x))|$, $b = \sup_{x\in \Omega}|f_z(y_1^{e}(x), y_2^{e}(x))|$, $c = \sup_{x\in \Omega}|g_y(y_1^{e}(x), y_2^{e}(x))|$ and $d = \sup_{x\in \Omega}|g_z(y_1^{e}(x), y_2^{e}(x))|$.

Let $A: = \Delta+aI$ where $D(A) = D(\Delta) = H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. It is clear that $A$ is a self-adjoint operator with a compact resolvent; hence, the spectrum of $A$ reduces to its point spectrum. More precisely, the eignvalues $(\lambda_{i})_{i\in \mathbb{N^*}}$ of $A$ are reals. We suppose that there exists a finite positive integer $N$ such that $\{\lambda_{i}\geq 0, \quad\forall i\in \{1, ..., N\}\}$ which is guaranteed thanks to the assumption $f_y(y_1^{e}, y_2^{e})\in L^{\infty}(\Omega)$.

Let us consider the following auxiliary linear system:

where $c_0 = \sup_{x\in \Omega}\left|d-\delta a\right|$. We mention that a simple application of the maximum principle (see ^[29]) gives that $0\leq z_1(t)\leq Z_1(t)$ and $0\leq z_2(t)\leq Z_2(t)$ for

Theorem 2.1. Let $B$ be a bounded operator on $H$; suppose that assumptions $(H_1)$, $(H_2)$ and $(H_3)$ hold; then, the feedback

where $D = (2\lambda+b+c+2\delta\lambda+2c_0)$, $\lambda = \max_{1\leq i \leq N}(\lambda_{i})$ and $\eta > 0$, ensures the exponential stabilization of system (2.3).

Remark 2.1. In the case where $B = I_d$, the feedback control $v(t)$ will be a constant, that is, $v(t) = -D-\eta$

Proof. On the one hand, from the first equation of (2.4), we have

then

On the other hand, from the second equation of (2.4), we have

then

Combining (2.6) and (2.7), we obtain

where $D = (2\lambda+b+c+2\delta\lambda+2c_0)$.

Using the expression of $v(t)$, we obtain

Integrating over $[k, k+1]$ for $k\in\mathbb{N^*}$, we obtain

then

By the recurrence argument, one can obtain

Since $\left\|(Z_1(k+1), Z_2(k+1))\right\|$ decreases then for $t\geq k$, we have

where $m = \ln(1+2\eta) > 0$. Recalling that $0\leq z_1(t)\leq Z_1(t)$ and $0\leq z_2(t)\leq Z_2(t)$ for

hence,

and therefore, there exists a positive constant $M_3$ such that

This completes the proof of Theorem 2.1.

2.2.   Nonlinear setting

In this section, we shall prove that if $\|(z^{0}_{1}, z^{0}_{2})\|_{\mathcal{H}}\leq\epsilon$ for $\epsilon$ small enough, then the local solution of (2.2) is global by using a Banach fixed point. Moreover, we show that this solution is exponentially stabilizable.

Theorem 2.2. Well-posedness

Let $B$ be a bounded operator on $H$, suppose that assumptions $(H_1)$, $(H_2)$ and $(H_3)$; then, for $(z^{0}_{1}, z^{0}_{2})\in \mathcal{H}$ such that $\|(z^{0}_{1}, z^{0}_{2})\|_{\mathcal{H}}\leq \epsilon$, where $\epsilon$ is sufficiently small, system $(2.2)$ admits a global solution $(z_1, z_2)\in L^{r}(0, \infty; \mathbb{H})$ for some $r\geq 1$.

Proof. System (2.2) can be written as

where

and

According to assumptions $(H_1)$ and $(H_2)$, we deduce that

for some positive integer $m$, where $r_i$ are such that $1\leq r_1 < \cdots < r_{m}\leq m$.

Let us consider

where $1\leq r\leq m$. We note by $(\Gamma(t, s))_{0 \leq s \leq t}$ the evolution system generated by $(z_1, z_2):\longrightarrow \mathcal{A}(z_1, z_2)+\mathcal{A}_0(z_1, z_2)+v(t)\mathcal{B}(z_1, z_2)$ (see Definition 5.3 p 126 ^[25]). Let the map

From (2.13), we conclude that $\Gamma(t, .)(z_1, z_2)\in L^{r}(0, \infty; \mathcal{H})$, that is

(see p 299 ^[12]). Moreover, using the superposition property of the evolution system $(\Gamma(t, s))_{0 \leq s \leq t}$, we deduce from (2.13) and (2.16) that there exists a positive constant $M_4$ such that

● We start by showing the invariance of $\Lambda$. We define

By duality arguments as in ^[31] page 197, we obtain

for all $(z_1, z_2)\in S(0, \rho)$.

Then

Substituting in (2.15), we obtain

and hence

Let us consider $\rho > 0$, which is chosen to satisfy the following constraints

and hence

Therefore

● Now we show that $\Lambda$ is a contraction map on $S(0, \rho)$.

Let us consider $(y_1, z_1), (y_2, z_2)\in S(0, r)$, then there exists a constant $C$ such that:

where the following argument is used

In the other hand, similar to (2.19), one can show as well that

Then $\Lambda$ is a contraction on $S(0, \rho)$ for $\rho$ chosen such that

then according to the Banach fixed point, system (2.2) has for $(z^{0}_{1}, z^{0}_{2})$ sufficiently small, a unique solution

Now we characterize the exponential stabilization of (2.2). Theorem 2.3 below is the main result of this paper.

Theorem 2.3. Let $B$ be a bounded operator on $H$, suppose that assumptions $(H_1)$, $(H_2)$ and $(H_3)$ hold, then the following feedback

where $D = 2\lambda+b+c+2\delta\lambda+2c_0$, $\lambda = \max_{1\leq i \leq N}(\lambda_{i})$, exponentially stabilizes (2.2).

Proof. We start by showing that the solution $(z_1, z_2)$ of (2.2) obeys the following estimate:

where

In fact, according to the variation of constants formula, we have

then

Using (2.19) in (2.28), we deduce that there exists a positive constant $C_3: = CM_4$ such that

then

where

Now, we prove the following lemma to achieve the proof of Theorem 2.3.

Lemma 2.1. There exist a time $T > 0$ and a constant $0 < \gamma < 1$ such that

Proof. According to the variation of constants formula (2.15), there exists a positive constant $K: = \sup_{s\in[0, T]}\|\Gamma(t, s)\|_{\mathcal{L}(\mathcal{H})}$ such that we have

then

where $h(\rho) = \alpha \rho^{m}$; we have

We can take $\rho = R > 0$ sufficiently small and $T$ sufficiently large so that

then, (2.30) yields.

We reiterate the argument obtaining

Let us consider $t_1 = nT$, then for all $t\geq t_1$ we conclude that

hence, there exist two positives constants $N$ and $\alpha_1$ such that

3.   Applications and numerical simulations

In this section, we apply the exponential stabilization result for the different Holling response functions Ⅰ, Ⅱ, Ⅲ and Ⅳ. We present numerical simulations for each example in two-dimensional space.

In the following, we consider the two-dimensional closed rectangular habitat $\Omega: = \{(x, y)/\:0\leq x \leq a, \: 0\leq y\leq b\}$. The eigenvalues and eigenfunctions respectively of the Laplacian operator $\Delta$ with the Neumann boundary in $\Omega$ are given by (see ^[16])

and

3.1.   Holling type Ⅰ

Let consider the following prey-predator-diffusion with a Holling type Ⅰ functional response.

where $f(y_1, y_2) = r_1y_1(1-y_1/\kappa_{1})-\beta y_1y_2\:$ and $g(y_1, y_2) = -r_2y_2+b\beta y_1y_2$.

Steady state solutions analysis:

System (3.3) has the following constant steady states

where $(y^*, z^*)$ is the solution of the following system

and $(y(x), z(x))$, where $(y(x), z(x))$ is a non-constant positive function when it exists. The asymptotic behavior of these steady states has been studied extensively; see, e.g., ^[6,11]. Furthermore, Kishimoto and Weinberger ^[18] showed that (3.3) has no stable positive steady-state solution when the domain $\Omega$ is convex, while, according to Theorem 2.3, these equilibrium states can be reached; more precisely, let $z_1 = y_1-y^{e}_1$ and $z_2 = y_2-y^{e}_2$; then, we have the following system:

where $BY = \frac{2}{1+\mu(x, y)}Y, \quad \forall Y\in L^2(\Omega)$ and $\mu\in L^{\infty}(\Omega)$ such that $\mu(x, y)\geq 1, \: \forall (x, y)\in \Omega$, is exponentially stabilizable for all equilibrium states $(y_1^{e}, y_2^{e})$.

Let us verify the conditions of Theorem 2.3

-Following ^[20], system (3.3) has a unique smooth non-negative solution for $y_{1}^{0}(x)\geq 0$ and $y_{2}^{0}(x)\geq 0$.

- By simple calculus, there exist positive constants $C_1, C_2, C_3, C_4, C_5$ and $C_6$ such that

and

then $(H_2)$ holds for $r_1 = 1$, $r_2 = 2$, $m_0 = 2$ and $C = \max\{C_1, C_2, C_3, C_4, C_5, C_6\}.$

Numerical simulations:

Let $\Omega = [0,300]\times[0,300]$, $B = I_d$, $r_1 = 1$, $\kappa_{1} = 1$, $e = 2.5$, $\beta = 0.4$, $\delta = 1$, $r_2 = 0.6$ and $b = 5$. The choice of parameters for numerical simulation was inspired from ^[22]. We illustrate numerically the exponential stabilization of (3.6) towards $(y^{e}_1, y^{e}_2) = (0, 0)$. To this end we start by the explicitation of the feedback control $v(t)$. On the one hand, by simple calculus we have:

On the other hand, the eigenvalues of $A: = \Delta+aI$ are:

It is clear that $A$ has a finite number of positive eigenvalues, the largest one being $\lambda_{0, 0}: = 1$. Then following Theorem 2.3, we obtain $v(t) = -5$. Let

and

Using the 2D finite difference (see ^[14]), we obtain Figure 1, which shows the densities of the uncontrolled system, and Figure 2, which shows the densities of the controlled system.

3.2.   Holling type Ⅱ

Let us consider the following prey-predator- diffusion with a Holling type Ⅱ functional response.

where $f(y_1, y_2) = r_1y_1(1-y_{1}/\kappa_{1})-\frac{\beta y_1y_{2}}{1+ey_1}\:$ and $g(y_1, y_2) = -r_2y_2+\frac{b\beta y_1y_{2}}{1+ey_1}$.

Steady state solutions analysis:

System (3.8) has the following constant steady states

where $(y^*, z^*)$ is the positive solution of the system

and $(y(x), z(x))$ where $(y(x), z(x))$ is a non-constant positive function when it exists. Camara and Aziz-Alaoui (see ^[8] and ^[7]) show that for suitable conditions on $r_1, \beta$ and $e$, system (3.8) has at least one positive solution. They have shown that $(0, 0)$ and $(\kappa_{1}, 0)$ are unstable, while, according to Theorem 2.3, these equilibrium states can be reached; more precisely, let $z_1 = y_1-y^{e}_1$, $z_2 = y_2-y^{e}_2$; then, we have the following system

where $BY = \mu(x, y)Y, \quad \forall Y\in L^2(\Omega)$ and $\mu\in L^{\infty}(\Omega)$ such that $\mu(x, y)\geq 1, \: \forall (x, y)\in \Omega$, is exponentially stabilizable for all equilibrium states $(y_1^{e}, y_2^{e})$. Let us verify the conditions of Theorem 2.3:

-Following Lemma 14.20 ^[27], system (3.8) has a non-negative solution for $y^{0}_{1}(x)\geq 0$ and $y^{0}_{2}(x)\geq 0$.

-By simple calculus, there exist positive constants $C_1, C_2, C_3$ and $C_4$ such that

and

then $(H_2)$ holds for $r_1 = 1$, $r_2 = 2$, $m_0 = 2$ and $C = \max\{C_1, C_2, C_3, C_4\}.$

Numerical simulations:

Let $\Omega = [0,500]\times[0,500]$, $B = I_d$, $r_1 = 1$, $\kappa_{1} = 1$, $e = 2.5$, $\beta = 0.4$, $\delta = 1$, $r_2 = 0.6$, $b = 5$. Let explicit the feedback control that stabilizes the solution of (3.11) towards $(y^{e}_1, y^{e}_2) = (0, 0)$. The eigenvalues of $A: = \Delta+aI$ are:

It is clear that $A$ has a finite number of positive eigenvalues, the largest one being $\lambda_{0, 0}: = 1$. By simple calculations we obtain

Let

and

Using the 2D finite difference (see ^[14]), we obtain Figure 3, which shows the densities of the uncontrolled system, and Figure 4, which shows the densities of the controlled system.

3.3.   Holling type Ⅲ

Let us consider the following prey-predator- diffusion with a Holling type Ⅲ functional response:

where $f(y_1, y_2) = r_1y_1(1-y_{1}/\kappa_{1})-\frac{\beta y_1^{2}y_{2}}{1+ey^{2}_1}$ and $g(y_1, y_2) = -r_2y_2+\frac{b\beta y_1^{2}y_{2}}{1+ey^{2}_1}.$

Steady state solutions analysis:

System (3.13) has the following constant steady states

where $(y^*, z^*)$ is the solution of the following system

Tian and Weng in ^[28] showed that $(y^*, z^*)$ exists and is positive for appropriate assumptions on $\beta, e, b$ and $r_2$, and they discussed the stability of this stationary solution. However, we have seen that, by using Theorem 2.3, these equilibrium states can be reached; more precisely, let $z_1 = y_1-y^{e}_1$, $z_2 = y_2-y^{e}_2$; then, we have then the following system

where $BY = (\mu(x, y)-1)(\mu(x, y)+1)Y, \quad \forall Y\in L^2(\Omega)$ and $\mu\in L^{\infty}(\Omega)$ such that $\mu(x, y)\geq 1, \: \forall (x, y)\in \Omega$, is exponentially stabilizable for all equilibrium states $(y^{1}_e, y^{2}_e)$. Now, let us verify the conditions of Theorem 2.3:

-Following Lemma 14.20 ^[27], system (3.13) has a non-negative solution for $y^{0}_{1}(x)\geq 0$ and $y^{0}_{2}(x)\geq 0$.

-By simple calculus, there exist positive constants $C_1, C_2, C_3$ and $C_4$ such that

and

then $(H_2)$ holds for $r_1 = 1$, $r_2 = 2$, $m_0 = 2$ and $C = \max\{C_1, C_2, C_3, C_4, C_5, C_6\}.$

Numerical simulations:

Let $\Omega = [0,350]\times[0,350]$, $B = I_d$, $r_1 = 1$, $\kappa_{1} = 1$, $e = 2.5$, $\beta = 0.4$, $\delta = 1$, $r_2 = 0.6$, $b = 5$. Let explicit the feedback control that stabilizes the solution of (3.16) towards $(y^{e}_1, y^{e}_2) = (0, 0)$. The eigenvalues of $A: = \Delta+aI$ are:

It is clear that $A$ has a finite number of positive eigenvalues, the largest one being $\lambda_{0, 0}: = 1$. By simple calculations we obtain

Let

and

Using the 2D finite difference (see ^[14]), we obtain Figure 5, which shows the densities of the uncontrolled system, and Figure 6, which shows the densities of the controlled system.

3.4.   Holling type Ⅳ

Let us consider the following prey-predator- diffusion with a Holling type Ⅳ functional response:

where $f(y_1, y_2) = r_1y_1(1-y_{1}/\kappa_{1})-\frac{\beta y_1^{2}y_{2}}{e_1+ey_1+e_2y^{2}_{1}}, \:$ $g(y_1, y_2) = -r_2y_2+\frac{b\beta y_1^{2}y_{2}}{e_1+ey_1+e_2y^{2}_{1}}$.

Steady state solutions analysis:

System (3.18) has the following constant steady states

where $(y^*, z^*)$ is the solution of the following system

We refer to ^[9] for discussions on the existence and stability of these equilibrium states. However, we have seen that, by using Theorem 2.3, these equilibrium states can be reached; more precisely, let $z_1 = y_1-y^{e}_1$, $z_2 = y_2-y^{e}_2$, then the following system

where $BY = (1-\mu(x))Y, \quad \forall Y\in L^{2}(\Omega)$ and $\mu\in L^{\infty}(\Omega)$, is exponentially stabilizable for all equilibrium states $(y^{1}_e, y^{2}_e)$.

Now, let us verify the conditions of Theorem 2.3:

-Following Lemma 14.20 ^[27], system (3.18) has a non-negative solution for $y^{0}_{1}(x)\geq 0$ and $y^{0}_{2}(x)\geq 0$.

-By simple calculus, there exist positive constants $C_1, C_2, C_3$ and $C_4$ such that

and

then $(H_2)$ holds for $r_1 = 1$, $r_2 = 2$, $m_0 = 2$ and $C = \max\{C_1, C_2, C_3, C_4\}.$

Numerical simulation:

Let $\Omega = [0,400]\times[0,400]$, $B = I_d$, $r_1 = 1$, $\kappa_{1} = 1$, $e = 0.7$; $e_1 = 0.5$, $e_2 = 2.5$, $\beta = 0.4$, $\delta = 1$, $r_2 = 0.6$ and $b = 5$. Let explicit the feedback control that stabilizes the solution of (3.21) towards $(y^{e}_1, y^{e}_2) = (0, 0)$. The eigenvalues of $A: = \Delta+aI$ are

It is clear that $A$ has a finite number of positive eigenvalues, the largest one being $\lambda_{0, 0}: = 1$. By simple calculations we obtain

Let

and

Using the 2D finite difference (see ^[14]), we obtain Figure 7, which shows the densities of the uncontrolled system, and Figure 8, which shows the densities of the controlled system.

4.   Conclusions

The problem of exponential stabilization of reaction-diffusion systems simulating predatory prey systems has been investigated. We constructed a multiplicative control that exponentially stabilizes the solution of the system to its equilibrium state. The designed controller has the advantage of reaching all equilibrium states. Numerical simulations show the efficiency of the used control.

Acknowledgments

This work was supported by the Ministry of Higher Education, Scientific Research and Innovation, the Digital Development Agency (DDA) and the National Center for Scientific and Technical Research (CNRST) under Project ALKHAWARIZMI/2020/19.

Conflict of interest

The authors declare that there is no conflict of interests regarding this paper.