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Asymptotic stability problem of predator–prey system with linear diffusion

Pubblicato online: 05 Sep 2022
Volume & Edizione: AHEAD OF PRINT
Pagine: -
Ricevuto: 09 Nov 2021
Accettato: 15 May 2022
Dettagli della rivista
License
Formato
Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Introduction

Since the beginning of the 21st century, human beings have experienced many major infectious diseases, including malaria and diseases caused by the severe acute respiratory syndrome (SARS) virus (atypical pneumonia) and Ebola virus. However, in early 2020, coronavirus disease-2019 (COVID-19) broke out and quickly raged all over the world.

After many outbreaks of infectious diseases, human beings began to study them in the hope of preventing the outbreak of the next epidemic. However, compared with nature, human ability is very limited. After many years of continuous struggle with infectious diseases, human understanding of some infectious diseases began to make some progress, and some good results were not achieved until the 20th century. In 1911, Dr Ross, a public health doctor, created a model using differential equations to study the dynamic behaviour of malaria transmission between mosquitoes and people. The results showed that if the number of mosquitoes can be controlled to no more than one value, the epidemic trend of malaria could be restrained. The study of stability and asymptotic stability plays an important role in the study of epidemic-related models.

Recently, a graph-theoretical approach has been developed [1,2,3,4,5,6,7,8], which systemises the construction of global Lyapunov functions of large-scale coupled systems. The approach has been successfully applied to resolve global-scale stability problems for the endemic equilibrium of multi-group epidemic models [3, 6].

In this paper, we utilise the graph-theory approach to investigate the stability of predator–prey models with preys travelling among n patches: {x˙i=xi(fi(xi)eiyi)+jinDij(xjαijxi),i=1,n.y˙i=yi(gi(yi)+εixi) \left\{ {\matrix{ {{{\dot x}_i} = {x_i}({f_i}({x_i}) - {e_i}{y_i}) + \sum\nolimits_{j \ne i}^n {{D_{ij}}({x_j} - {\alpha _{ij}}{x_i}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1, \cdots n.} } \hfill \cr {{{\dot y}_i} = {y_i}({g_i}({y_i}) + {\varepsilon _i}{x_i})} \hfill \cr } } \right. and another model: {x˙i=xi(fi(xi)eiyi)+j=1nDij(xjαijxi),i=1,n.y˙i=yi(gi(yi)+εixi)+j=1ndij(yjβijyi), \left\{ {\matrix{ {{{\dot x}_i} = {x_i}({f_i}({x_i}) - {e_i}{y_i}) + \sum\nolimits_{j = 1}^n {{D_{ij}}({x_j} - {\alpha _{ij}}{x_i}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1, \cdots n} .} \hfill \cr {{{\dot y}_i} = {y_i}({g_i}({y_i}) + {\varepsilon _i}{x_i}) + \sum\nolimits_{j = 1}^n {{d_{ij}}({y_j} - {\beta _{ij}}{y_i})} ,} \hfill \cr } } \right. Here, xi, yi represents the densities of the preys and predators on the patch i; Dij and dij are the dispersal rates of the preys and predators, respectively, from patch j to patch i; and constants αij, βij can be selected to represent different boundary conditions [9,10,11]. The parameters in Model (1) are non-negative, and ei, εi are positive. Much of the previous work related to Model (1) can be found elsewhere [10,11] and in references cited therein.

We want to find all equilibria by utilising the m-matrix, different from previous papers [6,8], and investigate the stability of some equilibria by utilising assumptions that are different from the literature [6,7,8].

In Section 3, we study the equilibria of Model (1). In Section 4, the asymptotic stability of some equilibria is demonstrated using Model (1). In Section 5, the existence and asymptotic stability of the equilibrium point of Model (2) are studied.

We always use the following notations: L=(i=2nD1iα1iD12D1nD21i2nD2iα2iD2nDn1Dn2i=1n1Dniαni)M=(f1(0)i=2nD1iα1iD12D1nD21f2(0)i2nD2iα2iD2nDn1Dn2fn(0)i=1n1Dniαni)M1=(f1(0)i=2nD1iα1i D1jD1,j+1D1nDk,1fk(0)ijnDkiαkiDknDk+1,1Dk+1,jfk+1(0)ek+1gk+11(0)ij+1n1Dk+1,iαk+1,i Dk+1,nDn,1Dn,jDn,j+1 fn(0)engn1(0)k=1n1Dniαni)M2=(f1(0)e1g1(0)i=2nD1iα1i D1jD1nDk,1fk(0)ekg1(0)ijnDkiαki DknDn,1Dn,j fn(0)engn1(0)k=1n1Dniαni) \matrix{ {L = \left( {\matrix{ { - \sum\limits_{i = 2}^n {D_{1i}}{\alpha _{1i}}} & {{D_{12}}} & \cdots & {{D_{1n}}} \cr {{D_{21}}} & { - \sum\limits_{i \ne 2}^n {D_{2i}}{\alpha _{2i}}} & {} & {{D_{2n}}} \cr {} & \cdots & \cdots & {} \cr {{D_{n1}}} & {{D_{n2}}} & \cdots & { - \sum\limits_{i = 1}^{n - 1} {D_{ni}}{\alpha _{ni}}} \cr } } \right)} \hfill \cr {M = \left( {\matrix{ {{f_1}(0) - \sum\limits_{i = 2}^n {D_{1i}}{\alpha _{1i}}} & {{D_{12}}} & \cdots & {{D_{1n}}} \cr {{D_{21}}} & {{f_2}(0) - \sum\limits_{i \ne 2}^n {D_{2i}}{\alpha _{2i}}} & {} & {{D_{2n}}} \cr {} & \cdots & \cdots & {} \cr {{D_{n1}}} & {{D_{n2}}} & \cdots & {{f_n}(0) - \sum\limits_{i = 1}^{n - 1} {D_{ni}}{\alpha _{ni}}} \cr } } \right)} \hfill \cr {{M_1} = \left( {\matrix{ {{f_1}(0) - \sum\limits_{i = 2}^n {D_{1i}}{\alpha _{1i}} } & \cdots & {{D_{1j}}} & {{D_{1,j + 1}}} & \cdots & {{D_{1n}}} \cr {} & \cdots & {} & \cdots & {} & {} \cr {{D_{k,1}}} & {} & {{f_k}(0) - \sum\limits_{i \ne j}^n {D_{ki}}{\alpha _{ki}}} & {} & {} & {{D_{kn}}} \cr {{D_{k + 1,1}}} & \cdots & {{D_{k + 1,j}}} & {\matrix{ {{f_{k + 1}}(0) - {e_{k + 1}}g_{k + 1}^{ - 1}(0)} \hfill \cr {\quad - \sum\limits_{i \ne j + 1}^{n - 1} {D_{k + 1,i}}{\alpha _{k + 1,i}}} \hfill \cr } } & {} & {{D_{k + 1,n}}} \cr {} & \cdots & {} & {} & {} & {} \cr {{D_{n,1}}} & \cdots & {{D_{n,j}}} & {{D_{n,j + 1}}} & \cdots & { {f_n}(0) - {e_n}g_n^{ - 1}(0) - \sum\limits_{k = 1}^{n - 1} {D_{ni}}{\alpha _{ni}}} \cr } } \right)} \hfill \cr {{M_2} = \left( {\matrix{ {{f_1}(0) - {e_1}{g^{ - 1}}(0) - \sum\limits_{i = 2}^n {D_{1i}}{\alpha _{1i}} } & { \cdots } & {{D_{1j}}} & \cdots & {{D_{1n}}} \cr {} & \cdots & {} & \cdots & {} \cr {{D_{k,1}}} & \cdots & {{f_k}(0) - {e_k}{g^{ - 1}}(0) - \sum\limits_{i \ne j}^n {D_{ki}}{\alpha _{ki}} } & \cdots & {{D_{kn}}} \cr {} & \cdots & {} & \cdots & {} \cr {{D_{n,1}}} & {} & {{D_{n,j}}} & {} & { {f_n}(0) - {e_n}g_n^{ - 1}(0) - \sum\limits_{k = 1}^{n - 1} {D_{ni}}{\alpha _{ni}}} \cr } } \right)} \hfill \cr } and s(M) denotes the maximum real part of all eigenvalues of matrix M. We assume that (H1)fi(0)>0,gi(0)=0,f˙i(xi)<0,g˙i(yi)<0,fori=1,,n;(H2)fi(0)eigi1(0)>0,f˙i(xi)eig˙i1(εixi)<0fori=k+1,,n;(H3)fi(0)eigi1(0)>0,f˙i(xi)eig˙i1(εixi)<0fori=1,,n;(H4)f˙i(xi)g˙i(yi)eiεi<0foranyxi,yi;(H5)fi(0)>0,gi(0)<0,f˙i(xi)<0,g˙i(yi)<0,fori=1,,n. \matrix{ {(H1){\kern 1pt} {\kern 1pt} {f_i}(0) > 0,{\kern 1pt} {\kern 1pt} {g_i}(0) = 0,{\kern 1pt} {\kern 1pt} {{\dot f}_i}({x_i}) < 0,{\kern 1pt} {\kern 1pt} {{\dot g}_i}({y_i}) < 0,{\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n;} \hfill \cr {(H2){\kern 1pt} {\kern 1pt} {f_i}(0) - {e_i}g_i^{ - 1}(0) > 0,{\kern 1pt} {\kern 1pt} {{\dot f}_i}({x_i}) - {e_i}\dot g_i^{ - 1}( - {\varepsilon _i}{x_i}) < 0{\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = k + 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n;} \hfill \cr {(H3){\kern 1pt} {\kern 1pt} {f_i}(0) - {e_i}g_i^{ - 1}(0) > 0,{\kern 1pt} {\kern 1pt} {{\dot f}_i}({x_i}) - {e_i}\dot g_i^{ - 1}( - {\varepsilon _i}{x_i}) < 0{\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,n;} \hfill \cr {(H4){\kern 1pt} {\kern 1pt} {{\dot f}_i}({x_i}){{\dot g}_i}({y_i}) - {e_i}{\varepsilon _i} < 0{\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} any{\kern 1pt} {\kern 1pt} {x_i},{\kern 1pt} {y_i};} \hfill \cr {(H5){\kern 1pt} {\kern 1pt} {f_i}(0) > 0,{\kern 1pt} {\kern 1pt} {g_i}(0) < 0,{\kern 1pt} {\kern 1pt} {{\dot f}_i}({x_i}) < 0,{\kern 1pt} {{\dot g}_i}({y_i}) < 0,{\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n.} \hfill \cr }

Existence of equilibria

We study the equilibria of Model (1). In order to find the equilibria of Model (1), we adopt the following equation set: {xi(fi(xi)eiyi)+jinDij(xjαijxi)=0i=1,nyi(gi(yi)+εixi)=0 \left\{ {\matrix{ {{x_i}({f_i}({x_i}) - {e_i}{y_i}){\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \sum\nolimits_{j \ne i}^n {{D_{ij}}({x_j} - {\alpha _{ij}}{x_i}) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots n} } \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {y_i}({g_i}({y_i}) + {\varepsilon _i}{x_i}) = 0} \hfill \cr } } \right.

Theorem 2.1

Model (1) always has a trial equilibrium E0 = (0; 0; ⋯ ; 0; 0).

Clearly, E0 = (0; 0; ⋯ ; 0; 0) is the zero solution of Model (2), which means that E0 = (0; 0; ⋯ ; 0; 0) is the trial equilibrium of Model (1).

Theorem 2.2

Model (1) has an equilibrium E1 = (x10; ;xn0;0) if the following conditions are satisfied:

L is irreducible;

(H1) is held;

s(M) > 0.

In fact, (x10; ⋯ ; xn0) is a positive equilibrium of xi=xifi(xi)+jinDij(xjαijxi),i=1,2n x_i^\prime = {x_i}{f_i}({x_i}) + \sum\limits_{j \ne i}^n {D_{ij}}({x_j} - {\alpha _{ij}}{x_i}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} 2{\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} n Here, the author compares the predator to a wolf and the prey to a sheep. This means that in all patchy environments, there are only sheep but no wolves (Fig. 1).

Fig. 1

Prey is analogous to sheep without predators.

Proof

Readers can read previous papers [6,7,8] for proof of the theorem.

Theorem 2.3

Model (1) has an equilibrium E2=(x10,0,,xko,0,xk+1*,yk+1*,xn*,yn*) {E_2} = ({x_{10}},{\kern 1pt} {\kern 1pt} 0,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {x_{ko}},{\kern 1pt} {\kern 1pt} 0,{\kern 1pt} {\kern 1pt} x_{k + 1}^*,{\kern 1pt} {\kern 1pt} y_{k + 1}^*, \cdots x_n^*,y_n^*) , if the following conditions are satisfied:

L is irreducible;

(H1) and (H2) are held;

s(M1) > 0

{xi=xifi(xi)+jinDij(xjαijxi)yi=0fori=1,2kxi=xi(fi(xi)eiyi)+jinDij(xjαijxi),yi=yi(gi(yi)+εixi)fori=k+1,,n \left\{ {\matrix{ {x_i^\prime = {x_i}{f_i}({x_i}) + \sum\nolimits_{j \ne i}^n {{D_{ij}}({x_j} - {\alpha _{ij}}{x_i})} } \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} y_i^\prime = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} 2{\kern 1pt} {\kern 1pt} \cdots k} \hfill \cr {x_i^\prime = {x_i}({f_i}({x_i}) - {e_i}{y_i}) + \sum\nolimits_{j \ne i}^n {{D_{ij}}({x_j} - {\alpha _{ij}}{x_i})} ,} \hfill \cr {y_i^\prime = {y_i}({g_i}({y_i}) + {\varepsilon _i}{x_i}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = k + 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n} \hfill \cr } } \right.

We know gi1 g_i^{ - 1} exists and gi1<0 g_i^{ - 1} < 0 by H(1). {xi=xifi(xi)+jinDij(xjαijxi)fori=1,2kxi=xi(fi(xi)eiyi)+jinDij(xjαijxi)fori=k+1,,n \left\{ {\matrix{ {x_i^\prime = {x_i}{f_i}({x_i}) + \sum\nolimits_{j \ne i}^n {{D_{ij}}({x_j} - {\alpha _{ij}}{x_i}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} 2{\kern 1pt} {\kern 1pt} \cdots k} } \hfill \cr {x_i^\prime = {x_i}({f_i}({x_i}) - {e_i}{y_i}) + \sum\nolimits_{j \ne i}^n {{D_{ij}}({x_j} - {\alpha _{ij}}{x_i}){\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = k + 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n} } \hfill \cr } } \right. Set Eq. (6) has a positive equilibrium (x10,0,,xk0,0,xk+1*,yk+1*,xn*,yn*) ({x_{10,}}{\kern 1pt} {\kern 1pt} 0,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {x_{k0}},{\kern 1pt} {\kern 1pt} 0,\;x_{k + 1}^*,y_{k + 1}^*,{\kern 1pt} {\kern 1pt} \cdots x_n^*,{\kern 1pt} {\kern 1pt} y_n^*) , if s(M1) > 0 and (H1) and (H2) are held [6,7,8]; therefore, Model (1) has an equilibrium: E2=(x10,0,,xk0,0,xk+1*,yk+1*,xn*,yn*). {E_2} = ({x_{10,}}{\kern 1pt} {\kern 1pt} 0,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {x_{k0}},{\kern 1pt} {\kern 1pt} 0,x_{k + 1}^*,{\kern 1pt} {\kern 1pt} y_{k + 1}^*,{\kern 1pt} {\kern 1pt} \cdots x_n^*,{\kern 1pt} {\kern 1pt} y_n^*). where yi*=gi1(εixi*) y_i^* = g_i^{ - 1}( - {\varepsilon _i}x_i^*) for i = k + 1, ⋯ , n. Readers may prove yi*>0 y_i^* > 0 by themselves [8].

This means that in some patchy environments, there are only sheep but no wolves. But in some patchy environments, sheep and wolves coexist (Fig. 2).

Fig. 2

Some patches have predators such as wolves.

Theorem 2.4

In Model (1), equilibrium does not exist. E2*=(0,y10,,0,yk0,xk+1*,yk+1*,xn*,yn*). E_2^* = (0,{y_{10}},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} 0,{\kern 1pt} {\kern 1pt} {y_{k0}},{\kern 1pt} {\kern 1pt} x_{k + 1}^*,{\kern 1pt} {\kern 1pt} y_{k + 1}^*,{\kern 1pt} {\kern 1pt} \cdots x_n^*,{\kern 1pt} {\kern 1pt} y_n^*). Readers may prove this theorem by themselves [8].

In some patchy environments, only wolves exist, but no sheep exist (Fig. 3).

Fig. 3

Some patches have no prey, but predators do not exist.

Theorem 2.5

In Model (1), equilibrium does not exist. E2*=(0,y10,,0,yn0). E_2^* = (0,{\kern 1pt} {\kern 1pt} {y_{10}},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} 0,{\kern 1pt} {\kern 1pt} {y_{n0}}). Readers may prove this theorem by themselves [8]. This indicates that in all patchy environments, only wolves but sheep do not exist (Fig. 4).

Fig. 4

State of all patches: they have no prey, but predators do not exist.

Theorem 2.6

Model (1) has an equilibrium E*=(x1*,y1*,,xn*,yn*) {E^*} = (x_1^*,{\kern 1pt} {\kern 1pt} y_1^*,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} x_n^*,{\kern 1pt} {\kern 1pt} y_n^*) , if the following conditions are satisfied:

L is irreducible;

(H1) and (H3) hold;

s(M2) > 0

Proof

Applying Theorem 2.2, we get the following form: xi=xi(fi(xi)eigi1(εixi))+jinDij(xiαijxi),fori=1,,n x_i^\prime = {x_i}({f_i}({x_i}) - {e_i}g_i^{ - 1}( - {\varepsilon _i}{x_i})) + \sum\limits_{j \ne i}^n {D_{ij}}({x_i} - {\alpha _{ij}}{x_i}),{\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n Equation set (7) has a positive equilibrium (x1*,y1*,xn*,yn*) (x_1^*,{\kern 1pt} {\kern 1pt} y_1^*,{\kern 1pt} {\kern 1pt} \cdots x_n^*,\;y_n^*) , by s(M2) > 0 and (H1) and (H3) [8]; therefore, Model (1) has an equilibrium E*=(x1*,y1*,xn*,yn*) {E^*} = (x_1^*,{\kern 1pt} {\kern 1pt} y_1^*,{\kern 1pt} {\kern 1pt} \cdots x_n^*,{\kern 1pt} {\kern 1pt} y_n^*) , where yi*=gi1(εixi*) y_i^* = g_i^{ - 1}( - {\varepsilon _i}x_i^*) .

This indicates that only wolves and sheep coexist in all patchy environments (Fig. 5).

Fig. 5

All patch predators coexist with preys.

Theorem 2.7

Suppose xi(0), yi(0) > 0 for i = 1,...,n. Then, Γ : {(x1, y1, ⋯ , xn,yn) ∈ R2n/xi, yi > 0, xi < xi0, yi < yi0} is positive invariance of Model (1).

Proof

First, we consider for all i, for all τ, xi(t), yi(t) > 0 in condition xi(0), yi(0) > 0.

Suppose there exist k > 0 and xi(k) = 0, xj(k) > 0; then, xi(k)=j=1nαijxj(k)>0 x_i^\prime (k) = \sum\nolimits_{j = 1}^n {\alpha _{ij}}{x_j}(k) > 0 .

Using similar steps, if yi > 0, then xi=xi(fi(xi)eiyi)+j=1nDij(xjαijxi)<xifi(xi)+j=1nDij(xjαijxi) x_i^\prime = {x_i}({f_i}({x_i}) - {e_i}{y_i}) + \sum\limits_{j = 1}^n {D_{ij}}({x_j} - {\alpha _{ij}}{x_i}) < {x_i}{f_i}({x_i}) + \sum\limits_{j = 1}^n {D_{ij}}({x_j} - {\alpha _{ij}}{x_i}) Suppose xi = xi0, xj < xj0, xi|xi=xi0<xi0fi(xi0)+j=1nDij(xj0αijxi0)=0 x_i^\prime {{\rm{|}}_{{x_i} = {x_{i0}}}} < {x_{i0}}{f_i}({x_{i0}}) + \sum\limits_{j = 1}^n {D_{ij}}({x_{j0}} - {\alpha _{ij}}{x_{i0}}) = 0 So, for all i, there exists xi0, xi < xi0. Using similar steps for all i, there exists yi0,yi<yi0=gi1(εixi0) {y_{i0}},{y_i} < {y_{i0}} = g_i^{ - 1}( - {\varepsilon _i}{x_{i0}}) .

Therefore, Γ : {(x1, y1, ⋯ , xn, yn) ∈ R2n/xi, yi > 0, xi < xi0, yi < yi0} is the positive invariance of Model (1), which means the uniform boundlessness of the solution in Γ/{E0, E1, E2, E*}.

Asymptotic stability of equilibria
Boundary equilibria
Theorem 3.1

If s(M) < 0 and (H1) holds, E0 is the asymptotic stability, and if s(M) > 0, E0 is unstable.

Readers can refer previous papers [6,7,8] for proof of the theorem.

Theorem 3.2

Suppose assumptions (H1) and (H4) hold, E1 is the asymptotic stability.

Proof

Denote the boundary equilibrium E1 = (x10,0,⋯ ,xn0,0) about Model (1), where xi0fi(xi0)+jinDij(xj0αijxi0)=0,fori=1,,n. {x_{i0}}{f_i}({x_{i0}}) + \sum\limits_{j \ne i}^n {D_{ij}}({x_{j0}} - {\alpha _{ij}}{x_{i0}}) = 0,{\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n. Consider a Lyapunov function Vi(xi,yi)=εi(xixi0+xi0lnxixi0)+eiyi {V_i}({x_i},{\kern 1pt} {\kern 1pt} {y_i}) = {\varepsilon _i}\left( {{x_i} - {x_{i0}} + {x_{i0}}\ln {{{x_i}} \over {{x_{i0}}}}} \right) + {e_i}{y_i} We show that Vi satisfies the assumption of Lemma 1: V˙i=εi(xixi0)x˙ixi)+eiy˙i=εif(ξ)(xixi0)22eiεi(xixi0)yi+eig(η)yi2+j=1nDijεixj0(xjxj0xixj0+1xjxi0xj0xi) \matrix{ {{{\dot V}_i}} \hfill & { = {\varepsilon _i}({x_i} - {x_{i0}}){{{{\dot x}_i}} \over {{x_i}}}) + {e_i}{{\dot y}_i}} \hfill \cr {} \hfill & { = {\varepsilon _i}{f^\prime }(\xi )({x_i} - {x_{i0}}{)^2} - 2{e_i}{\varepsilon _i}({x_i} - {x_{i0}}){y_i} + {e_i}{g^\prime }(\eta )y_i^2 + \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}{x_{j0}}\left( {{{{x_j}} \over {{x_{j0}}}} - {{{x_i}} \over {{x_{j0}}}} + 1 - {{{x_j}{x_{i0}}} \over {{x_{j0}}{x_i}}}} \right)} \hfill \cr } By assumptions (H1) and (H4), we get Vi<j=1nDijεixj0(1xjxi0xj0xi+lnxjxi0xj0xi)+j=1nDijεixj0[(xixi0+lnxixi0)(xjxj0lnxjxj0)]<j=1nDijεixj0[(Hi(xi)Hj(xi))] \matrix{ {V_i^\prime } \hfill & { < \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}{x_{j0}}\left( {1 - {{{x_j}{x_{i0}}} \over {{x_{j0}}{x_i}}} + {\rm{ln}}{{{x_j}{x_{i0}}} \over {{x_{j0}}{x_i}}}} \right) + \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}{x_{j0}}\left[ {\left( { - {{{x_i}} \over {{x_{i0}}}} + {\rm{ln}}{{{x_i}} \over {{x_{i0}}}}} \right) - \left( { - {{{x_j}} \over {{x_{j0}}}} - {\rm{ln}}{{{x_j}} \over {{x_{j0}}}}} \right)} \right]} \hfill \cr {} \hfill & { < \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}{x_{j0}}[({H_i}({x_i}) - {H_j}({x_i}))]} \hfill \cr } and Hi(xi) and Dij satisfy the assumptions of Lemmas 1 and 2 [8], then Vi<0 V_i^\prime < 0 .

Therefore, the function V(x1,y1,,xn,yn)=i=1nciVi(xi,yi),i=1,,n V({x_1},{\kern 1pt} {\kern 1pt} {y_1},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {x_n},{\kern 1pt} {\kern 1pt} {y_n}) = \sum\limits_{i = 1}^n {c_i}{V_i}({x_i},\;{y_i}),{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n as defined in Lemma 2 is a Lyapunov function for Model (1), and V˙<0 \dot V < 0 for all (x1, y1, ⋯ , xn, yn) ∈ R2n.

This also implies that E1 is unique, completing the proof of Theorem 3.2. We will discuss the stability of E2 in future.

Positive equilibrium

In this section, we prove that the positive equilibrium of Model (1) is the asymptotic stability if it exists.

Theorem 3.3

Suppose assumptions (H1) and (H4) hold; then, the positive equilibrium E*=(x1*,y1*,,xn*,yn*) {E^*} = (x_1^*,{\kern 1pt} {\kern 1pt} y_1^*,{\kern 1pt} {\kern 1pt} \cdots ,x_n^*,{\kern 1pt} {\kern 1pt} y_n^*) of System (1) is asymptotically stable.

Proof

First, we suppose that the positive equilibrium exists. We denote the positive equilibrium, for E*=(x1*,y1*,,xn*,yn*),xi*,yi*>0,fori=1,2n {E^*} = (x_1^*,{\kern 1pt} {\kern 1pt} y_1^*,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} x_n^*,{\kern 1pt} {\kern 1pt} y_n^*),{\kern 1pt} x_i^*,{\kern 1pt} {\kern 1pt} y_i^* > 0,{\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} 2{\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} n , about Model (1), whereby we establish a Lyapunov function Vi(xi,yi)=εi(xixi*+xi*lnxixi*)+ei(yiyi*+lnyiyi*)Vi=εi((xixi*)xixi)+ei(yiyi*)yi/yi=εif(ξ)(xixi*)22eiεi(xixi*)(yiyi*)+eig(η)(yiyi*)2+j=1nDijεixj*(xjxj*xixi*+1xjxi*xj*xi) \matrix{ {{V_i}({x_i},{y_i}) = {\varepsilon _i}\left( {{x_i} - x_i^* + x_i^*{\rm{ln}}{{{x_i}} \over {x_i^*}}} \right) + {e_i}\left( {{y_i} - y_i^* + {\rm{ln}}{{{y_i}} \over {y_i^*}}} \right)} \hfill \cr {\;\;\;\;\;\;\;\;\;V_i^\prime = {\varepsilon _i}\left( {\left( {{x_i} - x_i^*} \right){{x_i^\prime } \over {{x_i}}}} \right) + {e_i}({y_i} - y_i^*)y_i^\prime /{y_i}} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\; = {\varepsilon _i}{f^\prime }(\xi )({x_i} - x_i^*{)^2} - 2{e_i}{\varepsilon _i}({x_i} - x_i^*)({y_i} - y_i^*) + {e_i}{g^\prime }(\eta )({y_i} - y_i^*{)^2} + \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}x_j^*\left( {{{{x_j}} \over {x_j^*}} - {{{x_i}} \over {x_i^*}} + 1 - {{{x_j}x_i^*} \over {x_j^*{x_i}}}} \right)} \hfill \cr } By assumptions (H1) and (H4), we get Vi<j=1nDijεixj*(1xjxi*xj*xi+lnxjxi*xj*xi)+j=1nDijεixj*[(xixi*+lnxixi*)(xjxj*+lnxjxj*)]<j=1nDijεixj*(Gi(xi)Gj(xj)) \matrix{ {V_i^\prime } \hfill & { < \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}x_j^*\left( {1 - {{{x_j}x_i^*} \over {x_j^*{x_i}}} + {\rm{ln}}{{{x_j}x_i^*} \over {x_j^*{x_i}}}} \right) + \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}x_j^*\left[ {\left( { - {{{x_i}} \over {x_i^*}} + {\rm{ln}} - {{{x_i}} \over {x_i^*}}} \right) - \left( { - {{{x_j}} \over {x_j^*}} + {\rm{ln}}{{{x_j}} \over {x_j^*}}} \right)} \right]} \hfill \cr {} \hfill & { < \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}x_j^*({G_i}({x_i}) - {G_j}({x_j}))} \hfill \cr } and Gi(xi) and Dij satisfy the assumptions of Lemmas 1 and 2 [8]; then, V′ < 0.

Therefore, the function V(x1,y1,,xn,yn)=i=1nciVi(xi,yi),i=1,,n V({x_1},{\kern 1pt} {\kern 1pt} {y_1},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {x_n},{\kern 1pt} {\kern 1pt} {y_n}) = \sum\limits_{i = 1}^n {c_i}{V_i}({x_i},{\kern 1pt} {\kern 1pt} {y_i}),{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n as defined in Lemma 2 is a Lyapunov function for Model (1), and V′ < 0 for all (x1,y1,,xn,yn)R+2n. ({x_1},{\kern 1pt} {\kern 1pt} {y_1},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {x_n},{\kern 1pt} {\kern 1pt} {y_n}) \in R_ + ^{2n}. This implies that E* is asymptotically stable [9,10,11,12,13,14,15,16,17].

Remarks

Now let us consider Model (2): {x˙i=xi(fi(xi)eiyi)+j=1nDij(xjαijxi)fori=1,,ny˙i=yi(gi(yi)+εixi)+j=1ndij(yjβijyi) \left\{ {\matrix{ {{{\dot x}_i} = {x_i}({f_i}({x_i}) - {e_i}{y_i}) + \sum\nolimits_{j = 1}^n {{D_{ij}}({x_j} - {\alpha _{ij}}{x_i})} } \hfill \cr {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n} \hfill \cr {{{\dot y}_i} = {y_i}({g_i}({y_i}) + {\varepsilon _i}{x_i}) + \sum\nolimits_{j = 1}^n {{d_{ij}}({y_j} - {\beta _{ij}}{y_i})} } \hfill \cr } } \right.

Theorem 4.1

Model (2) always has a trial equilibrium E0 =(0, 0,⋯, 0, 0).

Theorem 4.2

Model (2) has an equilibrium E1 =(x10; 0; . . . ; xn0; 0) if the following conditions are satisfied:

L is irreducible;

(H5) holds;

s(M) > 0;

where (x10, ⋯, xn0) is a positive equilibrium of xifi(xi)+jinDij(xjαijxi)=0,i=1,2n {x_i}{f_i}({x_i}) + \sum\limits_{j \ne i}^n {D_{ij}}({x_j} - {\alpha _{ij}}{x_i}) = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} 2{\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} n

The proof is similar to that for Theorem 2.2.

The positive equilibrium E*=(x1*,y1*,xn*,yn*) {E^*} = (x_1^*,{\kern 1pt} {\kern 1pt} y_1^*,{\kern 1pt} {\kern 1pt} \cdots x_n^*,{\kern 1pt} {\kern 1pt} y_n^*) probably exists.

Theorem 4.3

Suppose xi(0) > 0, yi(0) > 0, for i = 1,⋯ ,n.

Γ : {((x1, y1, ⋯xn, yn) ∈ R2n/xi, yi > 0, xi < xi0, yi < yi0)} is the positive invariance of Model (2).

Proof

First, we consider for all i, for all t, xi(t), yi(t) > 0 according to condition xi(0), yi(0) > 0; suppose there exists k > 0 and xi(k) = 0, xj(k) > 0; then, xi(k)=j=1nαijxj(k)>0 x_i^\prime (k) = \sum\nolimits_{j = 1}^n {\alpha _{ij}}{x_j}(k) > 0 .

Using similar steps, yi > 0, then xi=(xi(fi(xi)eiyi)+j=1nDij(xjαijxi)<xifi(xi)+j=1nDij(xjαijxi) x_i^\prime = ({x_i}({f_i}({x_i}) - {e_i}{y_i}) + \sum\limits_{j = 1}^n {D_{ij}}({x_j} - {\alpha _{ij}}{x_i}) < {x_i}{f_i}({x_i}) + \sum\limits_{j = 1}^n {D_{ij}}({x_j} - {\alpha _{ij}}{x_i}) Suppose xi = xi0, xj, < xj0, xi|xi=xi0<xi0fi(xi0)+j=1nDij(xj0αijxi0)=0 x_i^\prime {{\rm{|}}_{{x_i} = {x_{i0}}}} < {x_{i0}}{f_i}({x_{i0}}) + \sum\limits_{j = 1}^n {D_{ij}}({x_{j0}} - {\alpha _{ij}}{x_{i0}}) = 0 So, for all i, there exists xi0, xi,< xi0.

Using similar steps for all i, there exists yi0, yi < yi0 εixi0<j=1ndijβijfori=1,,n {\varepsilon _i}{x_{i0}} < \sum\limits_{j = 1}^n {d_{ij}}{\beta _{ij}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n Therefore, Γ : {((x1,y1, ⋯xn,yn) ∈ R2n/xi, yi > 0, xi < xi0, yi < yi0)} is the positive invariance of Model (1), which means the uniform boundedness of the solution in Γ/{E0, E1}.

Theorem 4.4

If s(M) < 0 and (H5) holds, E0 is the asymptotic stability, and if s(M) > 0, E0 is unstable.

Readers can refer previous papers [6,7,8] for proof of the theorem.

Theorem 4.5

Suppose s(M) > 0 holds; if s(N) < 0, E1 is the asymptotic stability, and if s(N) > 0, E1 is unstable, where N=(ε1x10j=1nd1jβ1jd12d1nd21ε21x20j=1nd2jβ2jd2ndn1dn2εnxn0j=1n1dnjβnj) N = \left( {\matrix{ {{\varepsilon _1}{x_{10}} - \sum\nolimits_{j = 1}^n {{d_{1j}}{\beta _{1j}}} } & {{d_{12}}} & \cdots & {{d_{1n}}} \cr {{d_{21}}} & {{\varepsilon _{21}}{x_{20}} - \sum\nolimits_{j = 1}^n {{d_{2j}}{\beta _{2j}}} } & {} & {{d_{2n}}} \cr {} & \cdots & \cdots & {} \cr {{d_{n1}}} & {{d_{n2}}} & \cdots & {{\varepsilon _n}{x_{n0}} - \sum\nolimits_{j = 1}^{n - 1} {{d_{nj}}{\beta _{nj}}} } \cr } } \right)

Proof

First, let N = diag{εixi0} − F, where diag{εixi0}=(ε1x100000ε2x20000εnxi0) diag\{ {\varepsilon _i}{x_{i0}}\} = \left( {\matrix{ {{\varepsilon _1}{x_{10}}} & 0 & 0 & 0 \cr 0 & {{\varepsilon _2}{x_{20}}} & \cdots & 0 \cr {} & \cdots & \cdots & {} \cr 0 & 0 & \cdots & {{\varepsilon _n}{x_{i0}}} \cr } } \right) and F=(j=1nd1jβ1jd12d1nd21j=1nd2jβ2jd2ndn1dn2j=1n1dnjβnj) F = \left( {\matrix{ {\sum\nolimits_{j = 1}^n {{d_{1j}}{\beta _{1j}}} } & { - {d_{12}}} & \cdots & { - {d_{1n}}} \cr { - {d_{21}}} & {\sum\nolimits_{j = 1}^n {{d_{2j}}{\beta _{2j}}} } & {} & { - {d_{2n}}} \cr {} & \cdots & \cdots & {} \cr { - {d_{n1}}} & { - {d_{n2}}} & \cdots & {\sum\nolimits_{j = 1}^{n - 1} {{d_{nj}}{\beta _{nj}}} } \cr } } \right) Since diag{εixi0} is non-negative and F is a non-singular M-matrix, N = diag{εixi0} − F has a Z sign pattern [6, 8, 15]. s(N)<0ρ(diag1{εixi0}F)>1 s(N) < 0 \Leftrightarrow \rho (dia{g^{ - 1}}\{ {\varepsilon _i}{x_{i0}}\} F) > 1 Let (ω1, ⋯ , ωn) is the left eigenvalue of diag1{εixi0}F corresponding ρ(diag1{εixi0}F) > 1.

Since diag1{εixi0}F is irreducible, we know ωi > 0 [15]. Set V=i=1nωiεixi0yi V = \sum\limits_{i = 1}^n {{{\omega _i}} \over {{\varepsilon _i}{x_{i0}}}}{y_i} We obtain V=i=1nωiεixi0(yi(gi(yi)+εixi)+j=1ndij(yjβijyi)<i=1nωiεixi0(yi(gi(0)+εixi0)+j=1ndij(yiβijyi)<i=1nωiεixi0yi(εixi0)+j=1ndij(yiβijyi)=(ω1ε1x10,,ωnεnxn0)(diag{εixi0}F)(y1,yn)T=(ω1,,ωn)(1ρdiag1{εixi0}F)(y1,yn)T<0 \matrix{ {{V^\prime }} \hfill & { = \sum\limits_{i = 1}^n {{{\omega _i}} \over {{\varepsilon _i}{x_{i0}}}}({y_i}({g_i}({y_i}) + {\varepsilon _i}{x_i}) + \sum\limits_{j = 1}^n {d_{ij}}({y_j} - {\beta _{ij}}{y_i})} \hfill \cr {} \hfill & { < \sum\limits_{i = 1}^n {{{\omega _i}} \over {{\varepsilon _i}{x_{i0}}}}({y_i}({g_i}(0) + {\varepsilon _i}{x_{i0}}) + \sum\limits_{j = 1}^n {d_{ij}}({y_i} - {\beta _{ij}}{y_i})} \hfill \cr {} \hfill & { < \sum\limits_{i = 1}^n {{{\omega _i}} \over {{\varepsilon _i}{x_{i0}}}}{y_i}({\varepsilon _i}{x_{i0}}) + \sum\limits_{j = 1}^n {d_{ij}}({y_i} - {\beta _{ij}}{y_i})} \hfill \cr {} \hfill & { = \left( {{{{\omega _1}} \over {{\varepsilon _1}{x_{10}}}},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {{{\omega _n}} \over {{\varepsilon _n}{x_{n0}}}}} \right)(diag\{ {\varepsilon _i}{x_{i0}}\} - F)({y_1},{\kern 1pt} {\kern 1pt} \cdots {y_n}{)^T}} \hfill \cr {} \hfill & { = ({\omega _1},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {\omega _n})(1 - \rho dia{g^{ - 1}}\{ {\varepsilon _i}{x_{i0}}\} F)({y_1},{\kern 1pt} {\kern 1pt} \cdots {y_n}{)^T}} \hfill \cr {} \hfill & { < 0} \hfill \cr } and the equal sign holds if and only if yi = 0. Therefore, by LaSalle invariance principle [9,10,11,12,13], E1 is the global asymptotic stability.

Theorem 4.6

Suppose A and B are irreducible, s(M) > 0, s(N) > 0 and (H4) and (H5) hold; then, the positive equilibrium E*=(x1*,y1*,,xn*,yn*) {E^*} = (x_1^*,{\kern 1pt} {\kern 1pt} y_1^*,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} x_n^*,{\kern 1pt} {\kern 1pt} y_n^*) of Model (2) exists, and E* is unique and global asymptotically stable in the positive cone R+2n R_ + ^{2n} , if there exists λ > 0 such that Dijεixj*=λdijeiyj* {D_{ij}}{\varepsilon _i}x_j^* = \lambda {d_{ij}}{e_i}y_j^* . where A=(D11D12D1nD21D21D2nDn1Dn2Dnn)B=(d11d12d1nd21d21d2ndn1dn2dnn)L*=(i=2nd1iβ1id12d1nd21i2nd2iβ2id2ndn1dn2i=1n1dniβni) \matrix{ {A = \left( {\matrix{ {{D_{11}}} & {{D_{12}}} & \cdots & {{D_{1n}}} \cr {{D_{21}}} & {{D_{21}}} & \cdots & {{D_{2n}}} \cr {} & \cdots & \cdots & {} \cr {{D_{n1}}} & {{D_{n2}}} & \cdots & {{D_{nn}}} \cr } } \right)} \hfill \cr {B = \left( {\matrix{ {{d_{11}}} & {{d_{12}}} & \cdots & {{d_{1n}}} \cr {{d_{21}}} & {{d_{21}}} & \cdots & {{d_{2n}}} \cr {} & \cdots & \cdots & {} \cr {{d_{n1}}} & {{d_{n2}}} & \cdots & {{d_{nn}}} \cr } } \right)} \hfill \cr {{L^*} = \left( {\matrix{ {\sum\nolimits_{i = 2}^n {{d_{1i}}{\beta _{1i}}} } & { - {d_{12}}} & \cdots & { - {d_{1n}}} \cr { - {d_{21}}} & {\sum\nolimits_{i \ne 2}^n {{d_{2i}}{\beta _{2i}}} } & {} & { - {d_{2n}}} \cr {} & \cdots & \cdots & {} \cr { - {d_{n1}}} & { - {d_{n2}}} & \cdots & {\sum\nolimits_{i = 1}^{n - 1} {{d_{ni}}{\beta _{ni}}} } \cr } } \right)} \hfill \cr }

Proof

First, we prove that E0 and E1 are the only two boundary equilibria. We show that x¯i=0 {\bar x_i} = 0 for some i implies that x¯j=0 {\bar x_j} = 0 , for all j. If x¯i=0 {\bar x_i} = 0 for some i, then 0=x¯i=i=1nDijx¯j>0 0 = {\bar x_i} = \sum\limits_{i = 1}^n {D_{ij}}{\bar x_j} > 0

Therefore, if Dij > 0, then x¯j=0 {\bar x_j} = 0 . Using the irreducibility of L, we conclude that x¯i=0 {\bar x_i} = 0 for all i.

If x¯i=0 {\bar x_i} = 0 for ∀i, then (y¯1,,y¯n) ({\bar y_1},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {\bar y_n}) is an equilibrium of y˙i=yi(gi(yi))+j=1ndij(yjβijyi)fori=1,,n {\dot y_i} = {y_i}({g_i}({y_i})) + \sum\limits_{j = 1}^n {d_{ij}}({y_j} - {\beta _{ij}}{y_i}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n We know that Eq. (12) only has a unique equilibrium (0,⋯ ,0), since ρ{diag{−gi(0)}(L*)1} < 1. Therefore, if x¯i=0 {\bar x_i} = 0 for all i, then y¯i=0 {\bar y_i} = 0 for all i. If y¯i=0 {\bar y_i} = 0 for some i, see Theorem 2.2.

Therefore, Model (2) has only two boundary equilibria E0 and E1.

Because s(M) > 0, s(N) > 0, A and B are irreducible, E0 and E1 are unstable. Using a uniform persistence result from previous works [10,11,12,13,14], we show that when s(M) > 0, s(N) > 0, the instability of E0 and E1 implies the uniform persistence of Model (2). The uniform persistence of Model (2) and the uniform boundedness of solution in Γ/{E0, E1}, implies E* exists in Γ/{E0, E1}.

The paper by Shuai et al. [7] only supposes that the positive equilibrium exists.

We denote the positive equilibrium E*=(x1*,y1*,,xn*,yn*),x1*,y1*>0,fori=1,,n {E^*} = (x_1^*,{\kern 1pt} {\kern 1pt} y_1^*,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} x_n^*,{\kern 1pt} {\kern 1pt} y_n^*),{\kern 1pt} {\kern 1pt} x_1^*,{\kern 1pt} {\kern 1pt} y_1^* > 0,\;for\;i = 1, \cdots ,n about Model (2). where {xi*(fi(xi*)eiyi*)+j=1nDij(xj*αijxi*)=0i=1,nyi*(gi(yi*)+εixi*)+j=1ndij(yj*βijyi*)=0 \left\{ {\matrix{ {x_i^*({f_i}(x_i^*) - {e_i}y_i^*) + \sum\nolimits_{j = 1}^n {{D_{ij}}(x_j^* - {\alpha _{ij}}x_i^*) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots n} } \hfill \cr {y_i^*({g_i}(y_i^*) + {\varepsilon _i}x_i^*) + \sum\nolimits_{j = 1}^n {{d_{ij}}(y_j^* - {\beta _{ij}}y_i^*) = 0} } \hfill \cr } } \right. Consider a Lyapunov function in the paper by Shuai et al. [7] for a single patch predator–prey model: Vi(xi,yi)=εi(xixi*+xi*lnxixi*)+ei(yiyi*+yi*lnyiyi*)V˙i=εi(xixi*)x˙ixi*+ei(yiyi*)y˙iyi*=εif(ξ)(xixi*)22ei(xixi*)(yiyi*)+eig(η)(yiyi*)2 \matrix{ {{V_i}({x_i},{\kern 1pt} {\kern 1pt} {y_i}) = {\varepsilon _i}\left( {{x_i} - x_i^* + x_i^*{\rm{ln}}{{{x_i}} \over {x_i^*}}} \right) + {e_i}\left( {{y_i} - y_i^* + y_i^*{\rm{ln}}{{{y_i}} \over {y_i^*}}} \right)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;{{\dot V}_i} = {\varepsilon _i}({x_i} - x_i^*){{{{\dot x}_i}} \over {x_i^*}} + {e_i}({y_i} - y_i^*){{{{\dot y}_i}} \over {y_i^*}}} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\varepsilon _i}{f^\prime }(\xi )({x_i} - x_i^*{)^2} - 2{e_i}({x_i} - x_i^*)({y_i} - y_i^*) + {e_i}{g^\prime }(\eta )({y_i} - y_i^*{)^2}} \hfill \cr } +j=1nDijεixj*(xjxj*xixi*+1xjxi*xj*xi)+j=1ndijeiyj*(yjyj*yiyi*+1yjyi*yj*yi) \quad + \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}x_j^*\left( {{{{x_j}} \over {x_j^*}} - {{{x_i}} \over {x_i^*}} + 1 - {{{x_j}x_i^*} \over {x_j^*{x_i}}}} \right) + \sum\limits_{j = 1}^n {d_{ij}}{e_i}y_j^*\left( {{{{y_j}} \over {y_j^*}} - {{{y_i}} \over {y_i^*}} + 1 - {{{y_j}y_i^*} \over {y_j^*{y_i}}}} \right) Vi<j=1nDijεixj*(xixi*+lnxixi*λyiyi*+lnyiyi*)(xjxj*+lnxjxj*λyjyj*+lnyjyj*)=j=1nDijεixj*[Gi(xi,xi)Gj(xj,xj)] \matrix{ {V_i^\prime } \hfill & { < \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}x_j^*\left( { - {{{x_i}} \over {x_i^*}} + {\rm{ln}}{{{x_i}} \over {x_i^*}} - \lambda {{{y_i}} \over {y_i^*}} + {\rm{ln}}{{{y_i}} \over {y_i^*}}} \right) - \left( { - {{{x_j}} \over {x_j^*}} + {\rm{ln}}{{{x_j}} \over {x_j^*}} - \lambda {{{y_j}} \over {y_j^*}} + {\rm{ln}}{{{y_j}} \over {y_j^*}}} \right)} \hfill \cr {} \hfill & { = \sum\limits_{j = 1}^n {D_{ij}}{\varepsilon _i}x_j^*[{G_i}({x_i},{\kern 1pt} {x_i}) - {G_j}({x_j},{\kern 1pt} {x_j})]} \hfill \cr } and Gi(xi, yi) and Dij satisfy the assumptions of Lemmas 1 and 2 [12]; then, Vi<0 V_i^\prime < 0 .

Therefore, the function V(x1,y1,,xn,yn)=i=1nciVi(xi,yi),i=1,,n V({x_1},{\kern 1pt} {\kern 1pt} {y_1},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {x_n},{\kern 1pt} {\kern 1pt} {y_n}) = \sum\limits_{i = 1}^n {c_i}{V_i}({x_i},{\kern 1pt} {\kern 1pt} {y_i}),{\kern 1pt} {\kern 1pt} i = 1,{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} n as defined in Lemma 2 is a Lyapunov function for Model (2), and V′ < 0 for all (x1,y1,,xn,yn)R+2n. ({x_1},{\kern 1pt} {\kern 1pt} {y_1},{\kern 1pt} {\kern 1pt} \cdots ,{\kern 1pt} {\kern 1pt} {x_n},{y_n}) \in R_ + ^{2n}. This implies that E* is unique, completing the proof of Theorem 4.6.

Conclusions

In this paper, we establish a Lotka–Volterra dispersal predator–prey system in a patchy environment. We show the existence of the model boundary equilibria and asymptotic stability under an appropriate condition. The main methods studied in this paper are the method of global Lyapunov function and the results of graph theory. We also consider a predator–prey dynamical model in a patchy environment, wherein the prey and predator individuals in each compartment can travel among n patches. Our recent results on the predator–prey dynamical model have been applied to various ecological and epidemiological models: the sufficient conditions for the persistence of patch populations are obtained. The authors prove that the boundary equilibrium and positive equilibrium of the system are asymptotically stable under appropriate conditions. The results show that under appropriate conditions, the prey in each patch will not die out.

Fig. 1

Prey is analogous to sheep without predators.
Prey is analogous to sheep without predators.

Fig. 2

Some patches have predators such as wolves.
Some patches have predators such as wolves.

Fig. 3

Some patches have no prey, but predators do not exist.
Some patches have no prey, but predators do not exist.

Fig. 4

State of all patches: they have no prey, but predators do not exist.
State of all patches: they have no prey, but predators do not exist.

Fig. 5

All patch predators coexist with preys.
All patch predators coexist with preys.

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