1 Introduction

The investigation of relationships between organisms and their biotic and abiotic environments is central in describing and understanding ecological and biological processes. The issues raised require more and more a solicitation of tools from complex science theories [7, 8, 16, 38]. In particular, the issue of population interactions is to understand mechanisms impacting the dynamics of trophic food chain.

It is known that ecological communities generally display complex structures that are challenging to integrate into a realistic model [1, 20, 21, 25, 28, 40]. This problem has been considered from the theoretical point of view by Lotka [22–24], followed by Volterra [37]. Lotka has introduced the first model able to elucidate the main features of species interactions and populational cycles in predator–prey systems. However, to gain some more realistic and fundamental understanding of biodiversity conservation or ecosystem loss processes, it is important to take into account spatial degrees of freedom.

Ecosystem loss refers to the disappearance of an ecosystem, or an assemblage of organisms and the physical environment in which they exchange energy and matter. Many studies show the interactions between the effects of disturbances and habitat destruction [15, 30]. Although natural systems are heterogeneous, the direct influence of spatial heterogeneity on most ecological variables is unknown [17, 29, 45]. The lack of spatially explicit, large-scale field studies is identified as a major obstacle to the understanding of fundamental ecological processes. Moreover, the conditions of correlation between the spatial pattern shape and the degree of ecosystem quality are not yet well known.

Turing [35] showed how the coupling of reaction and diffusion can induce pattern formation. Since then, the mechanism responsible for the spontaneous generation of spatial patterns through biological or chemical interactions has been called diffusion instability. The classical Lotka–Volterra type systems are very important in the models of population dynamics. The governing equations for the population densities are described by a system of reaction-diffusion equations, for example, in food webs. A food web can be characterized by a model of complex network, in which a node represents a species Wang et al. [41–44] proposed a class of three-species Lotka–Volterra mutualism models with diffusion, time-varying delay, and linear coupling, which are described by systems of linear parabolic partial differential equations. Ryu and Ahn [31] studied the dynamics of predator–prey interaction systems between two species with ratio-dependent functional responses.

In this paper, we consider a system of reaction-diffusion equations to model the spatiotemporal dynamics of three species linked by a trophic food chain. Our model is based on a modified version of the Leslie–Gower scheme [18, 19]. Modified Leslie and Gower versions was studied by changing the functional response or by adding a third species in the trophic chain [6, 12, 36], also by incorporation a periodic impulsive perturbations [11]. We examine first the mathematical model analysis in Sect. 3 in which we explore the stability of the positive homogeneous equilibrium. Then by numerical simulation in Sect. 4, we focus on complex issues related to patterns of ecosystem loss. Through the analysis of bifurcations, pattern complexity is explored by determining the critical parameter values, which lead to ecosystem changes in a bottom-up scenario of control. We expect that understanding such effects, provide essential tools for the response prediction of communities to global changes or for ecosystem loss.

2 Materials and methods

2.1 Mathematical model

Originally, Leslie and Gower [18] have developed a temporal trophic model without incorporating the spatial dimension. Author in [12] analyzed in detail the nonspatial modified version, which uses the equations below:

$$ \left \{ \begin{array}{l} \displaystyle\frac{d E_1}{d T} = \displaystyle \biggl( a_0 - b_0 E_1 -\frac{v_0 E_2}{ E_1+d_0} \biggr) E_1 \\[12pt] \displaystyle\frac{d E_2}{d T} =\displaystyle \biggl( -a_1 +\frac{v_1 E_1}{ E_1+d_0} - \frac{v_2 E_3}{E_2+d_2} \biggr ) E_2,\\[12pt] \displaystyle\frac{d E_3}{d T} =\displaystyle \biggl( c_3 - \frac{v_3 E_3}{ E_2+d_3} \biggr) E_3, \end{array} \right . $$
(1)

with E 1(0)≥0, E 2(0)≥0, and E 3(0)≥0 are initial population densities.

E 1 represents the density of prey at time T. E 2 is the density of medium predator at time T and eating E 1. E 3 is the density of top predator at time T and eating E 2. a 0, b 0, v 0, d 0, a 1, v 1, v 2, d 2, c 3, v 3, and d 3 are model parameters assuming only positive values.

a 0, (respectively c 3) is the growth rate of prey E 1, (respectively top predator E 3). b 0 measures the strength of competition among individuals of species E 1. v 0 is the maximum value of the per capita reduction rate of E 1 due to E 2. v 1 is the maximum value of the per capita growth rate achieved by E 2. v 2 is the maximum value of the per capita reduction rate of E 2 due to E 3. v 3 is the maximum value that the decay rate that top predator E 3 can reach. d 0 measures the extent to which environment provides protection to prey E 1 and middle predator E 2. a 1 is the natural mortality rate of middle predator E 2. d 3 represents the residual loss, caused by high predation of E 2 top predator E 3.

In order to incorporate into (1) the spatial complexity of interactions, we naturally consider this associated reaction-diffusion system of equations,

$$ \left\{ \begin{array}{l} \displaystyle\frac{\partial U_1}{\partial T} = \displaystyle \biggl( a_0 - b_0 U_1 -\frac{v_0 U_2}{ U_1+d_0} \biggr) U_1 + \delta_1 \Delta U_1,\\[12pt] \quad T>0, \\[3pt] \displaystyle\frac{\partial U_2}{\partial T} =\displaystyle \biggl( -a_1 +\frac{v_1 U_1}{ U_1+d_0} - \frac{v_2 U_3}{U_2+d_2} \biggr ) U_2 \,{+}\, \delta_2 \Delta U_2,\\[12pt] \quad T>0,\\[3pt] \displaystyle\frac{\partial U_3}{\partial T} =\displaystyle \biggl( c_3 - \frac{v_3 U_3}{ U_2+d_3} \biggr) U_3 + \delta_3 \Delta U_3,\quad T>0 \end{array} \right. $$
(2)

where δ i is the diffusion rate of population i, (i=1,2,3) and Δ represent Laplacian operator in a bounded domain Ω 0 of \(\mathbb{R}^{n}\) (n=1,2,3). To investigate problem (2), we introduce the following scaling transformations:

We obtain the following equations:

$$ \left \{ \begin{array}{l} \displaystyle\frac{\partial W_1}{\partial t} = \displaystyle W_1(1-W_1) - \frac{ W_1 W_2}{W_1+a} + \varepsilon_1 \Delta W_1,\\[12pt] \quad t>0, \ x\in\varOmega\\[3pt] \displaystyle\frac{\partial W_2}{\partial t} = \displaystyle \frac{c W_1 W_2}{W_1+a} - bW_2 - \frac{W_2 W_3}{W_2 + d} + \varepsilon_2 \Delta W_2,\\[12pt] \quad t>0, \ x\in\varOmega\\[3pt] \displaystyle\frac{\partial W_3}{\partial t} = \displaystyle p W_3 - \frac{q W^2_3}{W_2 + r} + \varepsilon_3 \Delta W_3,\\[12pt] \quad t>0, \ x\in\varOmega \end{array} \right . $$
(3)

  • \(\frac{\partial W_{1}}{\partial\nu} = \frac {\partial W_{2}}{ \partial\nu} = \frac{\partial W_{3}}{\partial\nu} = 0\) on ∂Ω t>0

  • W 1(x,0)=W 01(x), W 2(x,0)=W 02(x), W 3(x,0)=W 03(x) x∈Ω

where Ω is a bounded domain in \(\mathbb{R}^{n}\), n=1,2,3, with smooth boundary ∂Ω and Δ is Laplacian operator. ε 1, ε 2, and ε 3 are respectively the diffusion rate of prey, medium predator, and top predator. \(\frac{\partial}{\partial\nu}\mid_{\partial\varOmega}\) is the outward normal of ∂Ω. Therefore, the three boundary conditions imply that the region is insular.

W 1(.,t), W 2(.,t), and W 3(.,t) are all densities, so only nonnegative solutions will make sense. In fact, this can be proved by using the comparison principle as long as the initial functions are nonnegative, [9, 33]. In order to simplify computations let us recall that without diffusion, the scaling system is written as follows:

$$ \left \{ \begin{array}{l} \displaystyle\frac{d E_1}{dt} = \displaystyle E_1(1-E_1) - \frac{ E_1 E_2}{E_1+a},\\[12pt] \displaystyle\frac{d E_2}{d t} = \displaystyle \frac{c E_1 E_2}{E_1+a} - bE_2 - \frac{E_2 E_3}{E_2 + d}, \\[12pt] \displaystyle\frac{d E_3}{d t}= \displaystyle p E_3 - \frac{q E^2_3}{E_2 + r}. \end{array} \right . $$
(4)

In the mathematical analysis section, we first study the permanence and persistence of solutions of systems (3), then we analyze the stability of the positive homogeneous equilibrium.

2.2 Numerical study

In the section numerical study section, we consider the equation system (2) with positive values of U 1(x,0), U 2(x,0), and U 3(x,0) in domain Ω=[0 100]×[0 100]. We set Neumann boundary conditions and fix the following parameter values: a 0=0.7, b 0=0.25, v 0=0.3, a 1=0.4, v 1=0.8, v 2=0.4, d 2=0.4, c 3=0.2, v 3=0.6, δ 1=0.04, δ 2=0.04, δ 3=0.002.

Note that in system (2) environmental prey carrying capacity K equals \(\frac{a_{0}}{b_{0}}\) and d_0 is its environmental protection. Thus, we define the percentage of protection (pp) by the following ratio: \(pp =\frac{d_{0}}{K}*100\). This index pp will be used as a measure of the environment quality in a bottom-up control. For a given value of pp, we can determine d_0 and have all the values system (2) parameters.

With the software Matlab R2009b, we solve the system (2) by implementing the combined finite differences method and the method Runge–Kutta 4. In our algorithm, we divide our domain Ω=[0 100]×[0 100] into 5002 small cells \(\displaystyle \omega_{ij}, \ i, j = 1,2, \ldots,500 \). At each time t, we compute the density U_l^ij(t) of each species l=1,2,3 in the cell ω_ij. Thus, we can quantify at time t, the spatial complexity sc(t) of each species distribution using the renormalized Shannon index [32]. For prey population, the spatial complexity sc(t) of the distribution at time t is

$$\mathrm{sc}(t)=-\frac{\sum_{i,j=1}^{500} ( \frac{U_1^{ij}(t)}{K}\ln ( \frac{U_1^{ij}(t)}{K} ) ) }{500^2+1} $$

The spatial complexity sc(t) ranges from 0 to 1. In fact, when sc(t)=0 is the completely ordered case equivalent to a uniform spatial distribution in all cells ω_ij. sc(t)=1 is the most complex case corresponding to the equal frequency of cells with densities varying from 0 to K.

For a given index value pp, we determine the evolution of the spatial complexity sc(t) over a period between 0 and T max=4500. Then we calculate the mean and standard deviation of the spatial complexity produced by this index pp on this observation period.

3 Mathematical results

3.1 Permanence and persistence of solutions

By standard existence theory, [2–4], it is not difficult to establish the local existence of the unique solution (W 1(.,t),W 2(.,t),W 3(.,t)) of (3) for 0=t<T max, where T max is determined by W 01(x), W 02(x), and W 03(x). Now we establish the global existence by proving that for any finite time T, \(\Vert W_{1}(., t)\Vert_{L^{\infty}}\), \(\Vert W_{2}(., t)\Vert_{L^{\infty}}\), \(\Vert W_{3}(., t)\Vert_{L^{\infty}}\) are bounded for 0≤t<T.

Theorem 1

For any smooth nonnegative functions W 01(x)≤1, W 02(x), and W 03(x), (3) has a unique smooth global solution for t>0.

Proof

First, it is easily seen that W 1(x,t)≥0, W 2(x,t)≥0, and W 3(x,t)≥0, since 0 is a sub-solution of each equation of (3). We have

$$ \left \{ \begin{array}{l} \displaystyle\frac{\partial W_1}{\partial t} \leq \displaystyle W_1(1-W_1) + \varepsilon_1 \Delta W_1\\[12pt] \displaystyle\frac{\partial W_1}{\partial\nu} = 0,\quad t>0,\\[12pt] \displaystyle W_1(x, 0) = W_{01}(x) \leq W_{01} \equiv \underset{\bar{\varOmega}}{\max} W_{01}(x),\quad x \in\varOmega. \end{array} \right . $$
(5)

By the comparison principle, we have W 1(x,t)≤W(t)≤1, where \(W(t) = e^{t} \frac{W_{01}}{1-W_{01}+e^{t}}\) is the solution of the initial value problem:

$$ \begin{array}{l} \left \{ \begin{array}{l} \displaystyle\frac{d W}{d t} =\displaystyle W(1-W),\\[12pt] \displaystyle W(0) =\displaystyle W_{01} \leq1, \end{array} \right .\\[24pt] \displaystyle\frac{\partial W_2}{\partial t} = \displaystyle\frac{c W_1 W_2}{W_1+a} - bW_2 - \frac{W_2 W_3}{W_2 + d} + \varepsilon_2 \Delta W_2. \end{array} $$
(6)

By the comparison principle, we deduce that

$$\frac{1}{c} \frac{\partial W_2}{\partial t} \leq \frac{1}{c} \frac{d E_2}{d t} $$

where E 2 is solution of system (4) with initial value \(E_{2}(0) = \max_{\bar{\varOmega}} W_{02}(x)\). Thus, from [5, 13],

$$\frac{1}{c} \frac{\partial W_2}{\partial t} \leq \frac{1}{c} \frac{d E_2}{d t} + \frac{d E_1}{d t}. $$

Let us denote by \(\sigma= \frac{1}{c}E_{2} + E_{1}\),

So,

$$W_2(x,t) \leq c + \frac{c}{4b},\quad \mbox{for}\ 0 \leq t < T. $$

From the third equation of system (3), we have

By the comparison principle, we deduce that

$$\alpha\frac{\partial W_3}{\partial t} \leq\alpha\frac{d E_3}{d t} $$

where E 3 is solution of system (4) with initial value \(E_{3}(0) = \max_{\bar{\varOmega}} W_{03}(x)\).

Thus,

So,

$$\alpha\frac{\partial W_3}{\partial t} +\alpha b W_3 \leq \frac{1}{4} + b + M $$

where \(\displaystyle M = \frac{1}{4q}\).

Finally,

$$W_3(x,t) \leq\frac{1}{\alpha} + \frac{1}{4\alpha b} + \frac{M}{\alpha b} \quad\mbox{for}\ 0 \leq t < T. $$

 □

Theorem 2

The domain

$$A \equiv [ 0,1 ] \times \biggl[0,c + \frac{c}{4b} \biggr]\times \biggl[0,\frac{1}{\alpha} + \frac{1}{4\alpha b} + \frac{M}{\alpha b} \biggr], $$

where

$$\alpha=\frac{4b}{(b+p)^2(4bc+c+4br)}, $$

is a positively invariant region for the global solutions of system (3).

Proof

As we can see before,

$$W_1 \leq E_1, \qquad W_2 \leq E_2,\qquad W_3 \leq E_3, $$

and from [5, 13] we have

 □

3.2 Local stability of the positive steady-state solutions

To investigate the stability of positive equilibria, we consider the solutions W_s(x) of system (3),

$$W_s(x) = \bigl( W_{1s}(x), W_{2s}(x), W_{3s}(x) \bigr), $$

satisfying \(( \frac{\partial W_{1}}{\partial t}, \frac {\partial W_{2}}{\partial t}, \frac{\partial W_{3}}{\partial t} )=(0,0,0) \). Let us consider a solution W(x,t) of system (3) having the following form:

(7)

Let us substitute W(x,t) with the form (7) into system (3) and then pick up all the terms which are linear in Z:

$$ \frac{\partial Z}{\partial t} = D \Delta Z + L(W_s) Z, $$
(8)

where

$$D = \mathrm{diag}(\varepsilon_1, \varepsilon_2, \varepsilon_3), $$
(9)

with

Theorem 3

A necessary condition for W s to be locally stable is, at W s ,

Proof

Let ϕ j denote the jth eigenfunction of −Δ on Ω with homogeneous Neumann boundary condition. That is,

(10)

for scalar λ j satisfying 0=λ 0<λ 1<λ 2<⋯.

We expand the solution Z of Eq. (8) via

$$ Z = \sum^{\infty}_{j = 0} z_j(t) \phi_j(x,y), $$
(11)

where each \(z_{j}(t) \in\mathbb{R}^{n}\). Using the form of Z given in (11) into Eq. (8) and equating the coefficients of each ϕ j , we have

$$\frac{d z_j}{d t} = C_j z_j, $$

where C j is the matrix

$$C_j = L(W_s) - \lambda_j D. $$

Now the solution W s is stable if and only if each z j (t) decays to zero. This is equivalent to the condition that each C j has three eigenvalues with negative real parts. The eigenvalues ρ k , k=1,2,3, of C j are determined by

$$\rho^3 + a \rho^2 + b \rho+ c = 0, $$

where

(12)

Now the three eigenvalues all have negative real part implies that

(13)
(14)
(15)

In particular, these three inequalities should hold for j=0.

For j=0, they are reduced to

(16)
(17)
(18)

 □

Recall the assumptions that guarantee the existence of positive steady states are

$$\begin{array}{l} F_{W_2} \leq0,\qquad F_{W_3} \leq0,\qquad G_{W_1} \geq0,\qquad G_{W_3} \leq0,\\[6pt] H_{W_1} \geq0, \qquad H_{W_2} \geq0. \end{array} $$

Under these conditions, by using the Routh–Hurwitz criteria (see [14]), we can obtain sufficient conditions for the stability of the positive steady state solutions. But it is too complicated for the general situation. If the positive steady state is constant, denoted by \((W^{*}_{1}, W^{*}_{2}, W^{*}_{3})\), we can prove the following theorem about the sufficient conditions.

Theorem 4

Let us assume that

$$p < 2,\quad b> \frac{c}{a} \quad \mbox{\textit{and}}\quad a \in \,\big] 7- 4\sqrt{3}, 7 + 4\sqrt{3} \big[ . $$

The homogeneous positive steady state solution \(W_{s} = (W^{*}_{1}, W^{*}_{2}, W^{*}_{3})\) is locally stable.

Proof

From the proof of Theorem 3, we know that we need only to prove that the solutions of

$$\rho^3 + a \rho^2 + b \rho+ c = 0, $$

have negative real parts, where a,b,c are given by (12). From the Routh–Hurwitz criteria, we know that it is true if

$$a > 0,\qquad ab-c > 0,\qquad abc - c^2 > 0. $$

To have this, we need only have a>0, ab>c>0.

From condition (12) , we need, for all j≥0,

$$ \left \{ \begin{array}{l} \lambda_j (\varepsilon_1 + \varepsilon_2 + \varepsilon_3) - (F_{W_1} + G_{W_2} + H_{W_3}) >0\\ \end{array} \right . $$
(19)
$$ \left \{ \begin{array}{l} \lambda^3_j \bigl(2 \varepsilon_1 \varepsilon_2 \varepsilon_3 + \varepsilon^2_1 \varepsilon_2 + \varepsilon^2_1 \varepsilon_3 + \varepsilon_1 \varepsilon^2_2 + \varepsilon_1 \varepsilon^2_3 + \varepsilon_3 \varepsilon^2_2 \\[3pt] \quad {}+ \varepsilon_2 \varepsilon^2_3\bigr)- \lambda^2_j\bigl(\varepsilon^2_1 G_{W_2} + \varepsilon^2_1 H_{W_3} + \varepsilon^2_2 F_{W_1}\\[3pt] \quad {} + \varepsilon^2_2 H_{W_3} + \varepsilon^2_3 F_{W_1} + \varepsilon^2_3 G_{W_2} + 2\varepsilon_1 \varepsilon_2 F_{W_1}\bigr)\\[3pt] \quad {}- \lambda^2_j( 2 \varepsilon_1 \varepsilon_2 H_{W_3} + 2 \varepsilon_1 \varepsilon_3 F_{W_1} + 2 \varepsilon_1 \varepsilon_3 G_{W_2} \\[3pt] \quad {}+ 2 \varepsilon_1 \varepsilon_2 G_{W_2}) - \lambda^2_j( 2 \varepsilon_1 \varepsilon_3 H_{W_3} + 2 \varepsilon_2 \varepsilon_3 F_{W_1}\\[3pt] \quad {}+ 2 \varepsilon_2 \varepsilon_3 G_{W_2} + 2 \varepsilon_2 \varepsilon_3 H_{W_3})+ \lambda_j \bigl(\varepsilon_1 G^2_{W_2}\\[3pt] \quad {} + \varepsilon_1 H^2_{W_3} + \varepsilon_2 F^2_{W_1} + \varepsilon_2 H^2_{W_3} + \varepsilon_3 F^2_{W_1}\\[3pt] \quad {} + \varepsilon_3 G^2_{W_2} + 2 \varepsilon_1 F_{W_1} G_{W_2} + 2 \varepsilon_1 F_{W_1} H_{W_3}\\[3pt] \quad {}+ 2 \varepsilon_1 G_{W_2} H_{W_3} - \varepsilon_1 F_{W_2} G_{W_1} - \varepsilon_1 F_{W_3} H_{W_1} \\[3pt] \quad {}+ 2 \varepsilon_2 F_{W_1} G_{W_2} + 2 \varepsilon_2 F_{W_1} H_{W_3} + 2 \varepsilon_2 G_{W_2} H_{W_3}\\[3pt] \quad {}- \varepsilon_2 F_{W_2} G_{W_1} - \varepsilon_3 G_{W_3} H_{W_2} + 2 \varepsilon_3 F_{W_2}u G_{W_2}v \\[3pt] \quad {} + 2 \varepsilon_3 F_{W_1} H_{W_3}w + 2 \varepsilon_3 G_{W_2} H_{W_3} - \varepsilon_3 F_{W_3} H_{W_1} \\[3pt] \quad {}- \varepsilon_3 G_{W_3} H_{W_2}\bigr) - F^2_{W_1} G_{W_2} - F^2_{W_1} H_{W_3} \\[3pt] \quad {}- 2F_{W_1} G_{W_2} H_{W_3} + F_{W_1} F_{W_2} G_{W_1} + F_{W_1} F_{W_3} H_{W_1} \\[3pt] \quad {}- F_{W_1} G^2_{W_2} - G^2_{W_2} H_{W_3} + F_{W_2} G_{W_1} G_{W_2}\\[3pt] \quad {}+ G_{W_2} G_{W_3} H_{W_2} + F_{W_2} G_{W_3} H_{W_1} + F_{W_3} G_{W_1} H_{W_2} \\[3pt] \quad {}- F_{W_1} H^2_{W_3} - G_{W_2} H^2_{W_3} + F_{W_3} H_{W_1} H_{W_3}\\[3pt] \quad {} + G_{W_3} H_{W_2} H_{W_3} > 0, \end{array} \right . $$
$$ \left \{ \begin{array}{l} \lambda^3_j \varepsilon_1 \varepsilon_2 \varepsilon_3 - \lambda^2_j (\varepsilon_1 \varepsilon_2 H_{W_3} + \varepsilon_1 \varepsilon_3 G_{W_2} + \varepsilon_2 \varepsilon_3 F_{W_1}) \\[6pt] \quad {} + \lambda_j( \varepsilon_1 G_{W_2} H_{W_3} + \varepsilon_2 F_{W_1} H_{W_3} + \varepsilon_3 F_{W_1} G_{W_2}\\[6pt] \quad {}- \varepsilon_1 G_{W_3} H_{W_2} - \varepsilon_3 F_{W_2} G_{W_1} - \varepsilon_2 F_{W_3} H_{W_1})\\[6pt] \quad {} + F_{W_1} G_{W_3} H_{W_2} + F_{W_2} G_{W_1} H_{W_3} + F_{W_3} G_{W_2} H_{W_1}\\[6pt] \quad {} - F_{W_1} G_{W_2} H_{W_3} - F_{W_2} G_{W_3} H_{W_1}\\[6pt] \quad {}- F_{W_3} G_{W_1} H_{W_2} > 0. \end{array} \right . $$

Observing that λ j ≥0, we need only

$$ \left \{ \begin{array}{l} F_{W_1} + G_{W_2} + H_{W_3} < 0, \\ \end{array} \right . $$
(20)
$$ \left \{ \begin{array}{l} \lambda^2_j\bigl(\varepsilon^2_1 G_{W_2} + \varepsilon^2_1 H_{W_3} + \varepsilon^2_2 F_{W_1} + \varepsilon^2_2 H_{W_3} + \varepsilon^2_3 F_{W_1}\\[6pt] \quad {} + \varepsilon^2_3 G_{W_2} + 2\varepsilon_1 \varepsilon_2 F_{W_1} + 2 \varepsilon_1 \varepsilon_2 H_{W_3} + 2 \varepsilon_1 \varepsilon_3 F_{W_1}\\[6pt] \quad {}+ 2 \varepsilon_1 \varepsilon_3 G_{W_2} + 2 \varepsilon_1 \varepsilon_2 G_{W_2}\\[6pt] \quad {} + 2 \varepsilon_1 \varepsilon_3 H_{W_3} + 2 \varepsilon_2 \varepsilon_3 F_{W_1}+ 2 \varepsilon_2 \varepsilon_3 G_{W_2}\\[6pt] \quad {}+ 2 \varepsilon_2 \varepsilon_3 H_{W_3}\bigr) - \lambda_j \bigl(\varepsilon_1 G^2_{W_2} + \varepsilon_1 H^2_{W_3} + \varepsilon_2 F^2_{W_1} \\[6pt] \quad {}+ \varepsilon_2 H^2_{W_3}+ \varepsilon_3 F^2_{W_1} + \varepsilon_3 G^2_{W_2} + 2 \varepsilon_1 F_{W_1} G_{W_2} \\[6pt] \quad {}+ 2 \varepsilon_1 F_{W_1} H_{W_3} + 2 \varepsilon_1 G_{W_2} H_{W_3} - \varepsilon_1 F_{W_2} G_{W_1} \\[6pt] \quad {}- \varepsilon_1 F_{W_3} H_{W_1} + 2 \varepsilon_2 F_{W_1} G_{W_2} + 2 \varepsilon_2 F_{W_1} H_{W_3} \\[6pt] \quad {}+ 2 \varepsilon_2 G_{W_2} H_{W_3} - \varepsilon_2 F_{W_2} G_{W_1} - \varepsilon_3 G_{W_3} H_{W_2} \\[6pt] \quad {}+ 2 \varepsilon_3 F_{W_2}u G_{W_2}v + 2 \varepsilon_3 F_{W_1} H_{W_3}w\\[6pt] \quad {}+ 2 \varepsilon_3 G_{W_2} H_{W_3} - \varepsilon_3 F_{W_3} H_{W_1}\\[5pt] \quad {}- \varepsilon_3 G_{W_3} H_{W_2}\bigr) + F^2_{W_1} G_{W_2} - F^2_{W_1} H_{W_3}\\[5pt] \quad {}- 2F_{W_1} G_{W_2} H_{W_3} + F_{W_1} F_{W_2} G_{W_1} + F_{W_1} F_{W_3} H_{W_1}\\[5pt] \quad {}- F_{W_1} G^2_{W_2} - G^2_{W_2} H_{W_3}+ F_{W_2} G_{W_1} G_{W_2} \\[5pt] \quad {}+ G_{W_2} G_{W_3} H_{W_2} + F_{W_2} G_{W_3} H_{W_1} + F_{W_3} G_{W_1} H_{W_2} \\[5pt] \quad {}- F_{W_1} H^2_{W_3}- G_{W_2} H^2_{W_3} + F_{W_3} H_{W_1} H_{W_3}\\[5pt] \quad {}+ G_{W_3} H_{W_2} H_{W_3} < 0, \end{array} \right . $$
(21)
$$ \left \{ \begin{array}{l} \lambda^2_j (\varepsilon_1 \varepsilon_2 H_{W_3} + \varepsilon_1 \varepsilon_3 G_{W_2} + \varepsilon_2 \varepsilon_3 F_{W_1})\\[5pt] \quad {}- \lambda_j( \varepsilon_1 G_{W_2} H_{W_3} + \varepsilon_2 F_{W_1} H_{W_3} + \varepsilon_3 F_{W_1} G_{W_2} \\[5pt] \quad {}- \varepsilon_1 G_{W_3} H_{W_2} - \varepsilon_3 F_{W_2} G_{W_1} - \varepsilon_2 F_{W_3} H_{W_1})\\[5pt] \quad {}- F_{W_1} G_{W_3} H_{W_2} - F_{W_2} G_{W_1} H_{W_3} - F_{W_3} G_{W_2} H_{W_1}\\[5pt] \quad {} + F_{W_1} G_{W_2} H_{W_3} + F_{W_2} G_{W_3} H_{W_1}\\[5pt] \quad {}+ F_{W_3} G_{W_1} H_{W_2} < 0. \end{array} \right . $$
(22)

At

and thanks to the hypothesis of the theorem,

and,

The second and third inequalities (21) and (22) hold because we have, \(F_{W_{1}} < 0\), \(F_{W_{2}} < 0\), \(F_{W_{3}} = 0\), \(G_{W_{1}} \geq1\), \(G_{W_{2}} < 0\), \(G_{W_{3}} < 0\), \(H_{W_{1}} = 0\), \(H_{W_{2}} > 0\), \(H_{W_{3}} < 0\).

This completes the proof of the theorem. □

3.3 Global stability of the positive steady-state solutions

Theorem 5

Let us assume that

(23)

Then, the homogeneous positive steady state solution \((W^{*}_{1}, W^{*}_{2}, W^{*}_{3})\) is globally asymptotically stable.

Proof

Let us denote by

(24)
(25)

We need to prove that Φ is a Lyapunov function, so we have to show that \(\displaystyle\frac{d\varPhi}{dt} < 0\).

(26)

We have

(27)

where

(28)

From hypothesis (23), the matrix B is positive, and consequently I 1<0.

On the other hand, we have

(29)

This completes the proof. □

4 Numerical results of far-from equilibrium dynamics

Figure 1(A) corresponding to a percentage of protection pp=5, shows the distribution of communities of prey, medium predator, and top predator in the domain Ω at time T max. Thus the system is far-from-equilibrium. Figures 1(B), (C), and (D) are obtained for pp=10, pp=15, and pp=20. Thus, variations in the percentage of protection pp impact significantly on the qualitative aspects of the pattern of different communities. The pattern structures are more ordered for lower values of pp than for higher values.

Fig. 1
figure 1

Spatial distribution of communities of prey, medium predator, and top predator at time t=4500, for different index values of environmental quality, in (A) pp=5, in (B) pp=10, in (C) pp=15, in (D) pp=20

Full size image

For a given percentage of protection pp, the pattern structure produced by each community is not static (see Fig. 2). It varies in time with a strong increase in the spatial complexity sc(t) at the beginning of observations and with small oscillation amplitudes beyond t=500 (see Fig. 2). In practice, several measures are therefore necessary for a better approximation of sc(t).

Fig. 2
figure 2

Dynamics of prey spatial complexity sc(t), for a given proportion of protection pp=10

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Qualitative differences of patterns (Fig. 1) induced by the change in pp are quantified in Fig. 3. Thus, the average value of the spatial complexity sc(t) is an increasing function of the proportion of protection index pp, with pp∈[1,20]. This average value of sc(t) in neighborhoods of equilibria decreases, before to vanish.

Fig. 3
figure 3

Mean and standard deviation of spatial complexity of prey pattern over environment quality

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5 Conclusions and discussions

In this study, we first perform the local stability analysis of the positive equilibrium homogeneous of our spatiotemporal system modeling three species interaction dynamics. Then we established the conditions for global stability of this positive equilibrium homogeneous by constructing a Lyapunov function. We show numerically that a low prey environmental protection produces light pressure on species leading to the extinction of prey population and medium predator population. This means the system converges to the homogeneous equilibrium \(( 0 , 0 , \frac{c_{3} d_{3}}{v_{3}} ) \). However, a high prey environmental protection makes the population of medium predator vulnerable and increases the risk of extinction. Therefore, the system converges to the homogeneous equilibrium \(( K , 0 , \frac{c_{3} d_{3}}{v_{3}} ) \). Through direct habitat changing, bottom-up effects lead to a cascade of fluctuations only for far-from equilibrium ecosystems. This helps to capture and quantify the differences between order and disorder. Moreover, this helps to establish the conditions of positive correlation between the spatial distribution complexity and the degree of environmental protection for species. However, the attractors of ecosystem equilibria alter the correlation between environmental quality and the spatial pattern of species. However, although our model is composed of three species, diversity is low and interspecies competition is not considered. These factors can be considered in the future.

Obviously, the measurement of pattern complexity is critical for understanding and preventing the ling bursting risk from environmental quality changes. This confirms the importance of the spatial dimension [10, 26, 27, 34, 39] in the study of ecosystem conservation and supports the work of dos Santos et al. [30], which state that predictions of the impacts of changing disturbance rates on biodiversity depend on community structure and cannot be made without knowledge of concurrent permanent habitat destruction. A detailed evaluation of ecosystem loss before restoration and a precise understanding of the factors structuring assemblages in the systems to be restored appear to be prerequisites [17]. Furthermore, these measurements can provide precision characterization and quantification for the spatial characteristics in biomonitoring approaches.