Multiple positive periodic solutions for a food-limited two-species Gilpin-Ayala competition patch system with periodic harvesting terms

Abstract

By using Mawhin’s coincidence degree theory and some inequality techniques, this paper establishes a new sufficient condition on the existence of at least eight positive periodic solutions for a food-limited two-species Gilpin-Ayala competition patch system with periodic harvesting terms. An example is given to illustrate the effectiveness of the result.

1 Introduction

In the past years, the study of population dynamics with harvesting in mathematical bioeconomics, due to its theoretical and practical significance in the optimal management of renewable resources, has attracted much attention [1–8]. Huusko and Hyvarinen in [9] pointed out that ‘the dynamics of exploited populations are clearly affected by recruitment and harvesting, and the changes in harvesting induced a tendency to generation cycling in the dynamics of a freshwater fish population.’ Recently, some researchers have paid much attention to the investigation of harvesting-induced multiple positive periodic solutions for some population systems under the assumption of periodicity of the parameters by using Mawhin’s coincidence degree theory [5–8]. In 1973, Gilpin and Ayala in [10] firstly proposed and studied a few Gilpin-Ayala type competition models. Since then, many papers have been published on the dynamics of Gilpin-Ayala type competition models (for example, see [11–15]).

In this paper, we consider a food-limited two-species Gilpin-Ayala competition patch system with harvesting terms:

$$\{\begin{array}{c}{x}_1^{\mathrm{\prime }}(t)=\frac{x_1(t)}{k_1(t)+c_1(t)x_1(t)}[a_1(t)-a_{11}(t)x_1^{{\theta }_1}(t)-a_{13}(t)y^{{\theta }_3}(t)]\hfill \\ \phantom{x_1^{\mathrm{\prime }}(t)=}+D_1(t)[x_2(t)-x_1(t)]-H_1(t),\hfill \\ x_2^{\mathrm{\prime }}(t)=\frac{x_2(t)}{k_2(t)+c_2(t)x_2(t)}[a_2(t)-a_{22}(t)x_2^{{\theta }_2}(t)]+D_2(t)[x_1(t)-x_2(t)]-H_2(t),\hfill \\ y^{\mathrm{\prime }}(t)=\frac{y(t)}{k_3(t)+c_3(t)y(t)}[a_3(t)-a_{33}(t)y^{{\theta }_3}(t)-a_{31}(t)x_1^{{\theta }_1}(t)]-H_3(t),\hfill \end{array}$$

(1.1)

where $x_1$ and y are the population densities of species x and y in patch 1, and $x_2$ is the density of species x in patch 2. Species y is confined to patch 1, while species x can diffuse between two patches due to the spatial heterogeneity and unbalanced food resources. $D_i(t)$ ($i=1,2$) are diffusion coefficients of species x. $a_1(t)$ ($a_2(t)$) is the natural growth rate of species x in patch 1 (patch 2), $a_3(t)$ is the natural growth rate of species y, $a_{13}(t)$, $a_{31}(t)$ are the inter-species competition coefficients. $a_{ii}(t)$ ($i=1,2,3$) are the density-dependent coefficients. $k_i(t)$ ($i=1,2$) are the population numbers of species x at saturation in patch 1 (patch 2), and $k_3(t)$ is the population number of species y at saturation in patch 1, respectively. $H_i(t)$ ($i=1,2,3$) denote the harvesting rates. ${\theta }_i$ ($i=1,2,3$) represent a nonlinear measure of interspecific interference. When $c_i(t)\ne 0$ ($i=1,2,3$), $\frac{a_i(t)}{k_i(t)c_i(t)}$ ($i=1,2,3$) are the rate of replacement of mass in the population at saturation (including the replacement of metabolic loss and of dead organisms). In this case, system (1.1) is a food-limited population model. For other food-limited population models, we refer to [16–19].

To our knowledge, few papers have been published on the existence of multiple positive periodic solutions for Gilpin-Ayala type competition patch models. Motivated by the work of Chen [20], we study the existence of multiple positive periodic solutions of (1.1) by using Mawhin’s coincidence degree theory. Since system (1.1) involves the diffusion terms, the rates of replacement and the interspecific interference, the methods used in [5–8] are not available to system (1.1).

2 Existence of multiple positive periodic solutions

For the sake of convenience and simplicity, we denote

$$\overline{g}=\frac{1}{T}{\int }_0^{T}g(t)dt,g^l=\underset{t\in [0,T]}{min}g(t),g^u=\underset{t\in [0,T]}{max}g(t),$$

where g is a nonnegative continuous T-periodic function.

Set

$$N_1=max\{{\left[{\left(\frac{a_1}{a_{11}}\right)}^u\right]}^{1/{\theta }_1},{\left[{\left(\frac{a_2}{a_{22}}\right)}^u\right]}^{1/{\theta }_2}\},N_2={\left[{\left(\frac{a_3}{a_{33}}\right)}^u\right]}^{1/{\theta }_3}.$$

From now on, we always assume that

(H₁) $k_i(t)$, $a_i(t)$, $a_{ii}(t)$, $H_i(t)$, $c_i(t)$ ($i=1,2,3$), $a_{13}(t)$, $a_{31}(t)$, $D_i(t)$ ($i=1,2$) are positive continuous T-periodic functions. ${\theta }_i$ ($i=1,2,3$) are positive constants.

(H₂) $\frac{k_1^l}{k_1^l+c_1^u{N}_1}{(\frac{a_1}{k_1})}^l>{(\frac{a_{13}}{k_1})}^u{(\frac{a_3}{a_{33}})}^u+D_1^u+(1+{\theta }_1){[{(\frac{a_{11}}{k_1})}^u]}^{\frac{1}{1+{\theta }_1}}{[\frac{H_1^u}{{\theta }_1}]}^{\frac{{\theta }_1}{1+{\theta }_1}}$.

(H₃) $\frac{k_2^l}{k_2^l+c_2^u{N}_1}{(\frac{a_2}{k_2})}^l>D_2^u+(1+{\theta }_2){[{(\frac{a_{22}}{k_2})}^u]}^{\frac{1}{1+{\theta }_2}}{[\frac{H_2^u}{{\theta }_2}]}^{\frac{{\theta }_2}{1+{\theta }_2}}$.

(H₄) $\frac{k_3^l}{k_3^l+c_3^u{N}_2}{(\frac{a_3}{k_3})}^l>{(\frac{a_{31}}{k_3})}^u{N}_1^{{\theta }_1}+(1+{\theta }_3){[{(\frac{a_{33}}{k_3})}^u]}^{\frac{1}{1+{\theta }_3}}{[\frac{H_3^u}{{\theta }_3}]}^{\frac{{\theta }_3}{1+{\theta }_3}}$.

(H₅) $H_i^l>D_i^u{N}_1$ ($i=1,2$).

We first make the following preparations [21].

Let X, Z be normed vector spaces, $L:domL\subset X\to Z$ be a linear mapping, $N:X\times [0,1]\to Z$ be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if $dimKerL=codimImL<+\mathrm{\infty }$ and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors $P:X\to X$ and $Q:Z\to Z$ such that $ImP=KerL$, $ImL=KerQ=Im(I-Q)$. If we define $L_P:domL\cap KerP\to ImL$ as the restriction $L{|}_{domL\cap KerP}$ of L to $domL\cap KerP$, then $L_P$ is invertible. We denote the inverse of that map by $K_P$. If Ω is an open bounded subset of X, the mapping N will be called L-compact on $\overline{\mathrm{\Omega }}\times [0,1]$ if $QN(\overline{\mathrm{\Omega }}\times [0,1])$ is bounded and $K_P(I-Q)N:\overline{\mathrm{\Omega }}\times [0,1]\to X$ is compact, i.e., continuous and such that $K_P(I-Q)N(\overline{\mathrm{\Omega }}\times [0,1])$ is relatively compact. Since ImQ is isomorphic to KerL, there exists an isomorphism $J:ImQ\to KerL$.

For convenience, we introduce Mawhin’s continuation theorem [[21], p.29] as follows.

Lemma 2.1 Let L be a Fredholm mapping of index zero and let $N:\overline{\mathrm{\Omega }}\times [0,1]\to Z$ be L-compact on $\overline{\mathrm{\Omega }}\times [0,1]$. Suppose

• (a) $Lu\ne \lambda N(u,\lambda )$ for every $u\in domL\cap \partial \mathrm{\Omega }$ and every $\lambda \in (0,1)$;

• (b) $QN(u,0)\ne 0$ for every $u\in \partial \mathrm{\Omega }\cap KerL$;

• (c) Brouwer degree ${deg}_B(JQN(\cdot ,0){|}_{KerL},\mathrm{\Omega }\cap KerL,0)\ne 0$.

Then $Lu=N(u,1)$ has at least one solution in $domL\cap \overline{\mathrm{\Omega }}$.

Set

$$h(x)=b-a{x}^{\alpha }-\frac{c}{x},x\in (0,+\mathrm{\infty }).$$

Lemma 2.2 Assume that a, b, c, α are positive constants and

$$b>(1+\alpha )a^{\frac{1}{1+\alpha }}{\left(\frac{c}{\alpha }\right)}^{\frac{\alpha }{1+\alpha }}.$$

Then there exist $0<x^{-}<x^{+}$ such that

Proof Since

$$h^{\prime }(x)=-a\alpha x^{\alpha -1}+\frac{c}{x^2}=0,x\in (0,+\mathrm{\infty })$$

implies that

$$x={\left(\frac{c}{a\alpha }\right)}^{\frac{1}{1+\alpha }},$$

we have

$$\underset{x\in (0,+\mathrm{\infty })}{sup}h(x)=b-a{\left(\frac{c}{a\alpha }\right)}^{\frac{\alpha }{1+\alpha }}-\frac{c}{{(\frac{c}{a\alpha })}^{\frac{1}{1+\alpha }}}=b-(1+\alpha )a^{\frac{1}{1+\alpha }}{\left(\frac{c}{\alpha }\right)}^{\frac{\alpha }{1+\alpha }}>0.$$

From this, it is easy to see that the assertion holds.

Set

 □

Lemma 2.3 Assume that (H₁)-(H₅) hold. Then the following assertions hold:

• (1) There exist $0<u_i^{-}<u_i^{+}$ such that

  $$M_i\left(u_i^{-}\right)=M_i\left(u_i^{+}\right)=0,$$

and

$$M_i(x)>0\mathit{\text{for}}x\in (u_i^{-},u_i^{+}),M_i(x)<0\mathit{\text{for}}x\in (0,u_i^{-})\cup (u_i^{+},+\mathrm{\infty }),i=1,2,3.$$

• (2) There exist $0<x_i^{-}<x_i^{+}$ such that

  $$p_i\left(x_i^{-}\right)=p_i\left(x_i^{+}\right)=0,$$

and

$$p_i(x)>0\mathit{\text{for}}x\in (x_i^{-},x_i^{+}),p_i(x)<0\mathit{\text{for}}x\in (0,x_i^{-})\cup (x_i^{+},+\mathrm{\infty }),i=1,2,3.$$

• (3) There exist $0<l_i^{-}<l_i^{+}$ such that

  $$m_i\left(l_i^{-}\right)=m_i\left(l_i^{+}\right)=0,$$

and

$$m_i(x)>0\mathit{\text{for}}x\in (l_i^{-},l_i^{+}),m_i(x)<0\mathit{\text{for}}x\in (0,l_i^{-})\cup (l_i^{+},+\mathrm{\infty }),i=1,2,3.$$

(4)

$$l_i^{-}<x_i^{-}<u_i^{-}<u_i^{+}<x_i^{+}<l_i^{+},i=1,2,3.$$

(2.1)

Proof It follows from (H₁)-(H₅) and Lemma 2.2 that the assertions (1)-(3) hold. Noticing that

we have

$$m_i(x)<p_i(x)<M_i(x),i=1,2,3.$$

It follows from this and the assertions (1)-(3) that the assertion (4) also holds. □

Lemma 2.4 [22]

Assume that $x\ge 0$, $y\ge 0$, $p>1$, $q>1$, and $\frac{1}{p}+\frac{1}{q}=1$. Then the following inequality holds:

$$x^{\frac{1}{p}}{y}^{\frac{1}{q}}\le \frac{x}{p}+\frac{y}{q}.$$

Now, we are ready to state the following main result of this paper.

Theorem 2.1 Assume that (H₁)-(H₅) hold. Then system (1.1) has at least eight positive T-periodic solutions.

Proof Since we are concerned with positive solutions of (1.1), we make the change of variables

$$x_j(t)=e^{u_j(t)}(j=1,2),y(t)=e^{u_3(t)}.$$

Then (1.1) is rewritten as

$$\{\begin{array}{c}{u}_1^{\mathrm{\prime }}(t)=\frac{1}{k_1(t)+c_1(t)e^{u_1(t)}}[a_1(t)-a_{11}(t)e^{{\theta }_1{u}_1(t)}-a_{13}(t)e^{{\theta }_3{u}_3(t)}]\hfill \\ \phantom{u_1^{\mathrm{\prime }}(t)=}+D_1(t)[\frac{e^{u_2(t)}}{e^{u_1(t)}}-1]-\frac{H_1(t)}{e^{u_1(t)}},\hfill \\ u_2^{\mathrm{\prime }}(t)=\frac{1}{k_2(t)+c_2(t)e^{u_2(t)}}[a_2(t)-a_{22}(t)e^{{\theta }_2{u}_2(t)}]+D_2(t)[\frac{e^{u_1(t)}}{e^{u_2(t)}}-1]-\frac{H_2(t)}{e^{u_2(t)}},\hfill \\ u_3^{\mathrm{\prime }}(t)=\frac{1}{k_3(t)+c_3(t)e^{u_3(t)}}[a_3(t)-a_{33}(t)e^{{\theta }_3{u}_3(t)}-a_{31}(t)e^{{\theta }_1{u}_1(t)}]-\frac{H_3(t)}{e^{u_3(t)}}.\hfill \end{array}$$

(2.2)

Take

$$X=Z=\{u={(u_1,u_2,u_3)}^T\in C(R,R^3):u_i(t+T)=u_i(t),i=1,2,3\}$$

and define

$$\parallel u\parallel =\underset{t\in [0,T]}{max}|u_1(t)|+\underset{t\in [0,T]}{max}|u_2(t)|+\underset{t\in [0,T]}{max}|u_3(t)|,u={(u_1,u_2,u_3)}^T\in X\text{ or }Z.$$

Equipped with the above norm $\parallel \cdot \parallel$, it is easy to verify that X and Z are Banach spaces.

Set

For any $u\in X$, because of the periodicity, we can easily check that ${\mathrm{\Delta }}_i(u,t,\lambda )\in C(R^2,R)$ ($i=1,2,3$) are T-periodic in t.

Let

Here, for any $k\in R^3$, we also identify it as the constant function in X or Z with the constant value k. It is easy to see that

$$KerL=R^3,ImL=\{u\in X:{\int }_0^T{u}_i(t)dt=0,i=1,2,3\}$$

is closed in Z, $dimKerL=codimImL=3$, and P, Q are continuous projectors such that

$$ImP=KerL,ImL=KerQ=Im(I-Q).$$

Therefore, L is a Fredholm mapping of index zero. On the other hand, $K_p:ImL\mapsto domL\cap KerP$ has the form

$$K_p(u)={\int }_0^{t}u(s)ds-\frac{1}{T}{\int }_0^T{\int }_0^{t}u(s)dsdt.$$

Thus,

where

$$\begin{array}{rcl}{\mathrm{\Phi }}_j(u,t,\lambda )& =& {\int }_0^t{\mathrm{\Delta }}_j(u,s,\lambda )ds-\frac{1}{T}{\int }_0^T{\int }_0^t{\mathrm{\Delta }}_j(u,s,\lambda )dsdt\\ -(\frac{t}{T}-\frac{1}{2}){\int }_0^T{\mathrm{\Delta }}_j(u,s,\lambda )ds,j=1,2,3.\end{array}$$

Obviously, QN and $K_p(I-Q)N$ are continuous. By the Arzela-Ascoli theorem, it is not difficult to show that $\overline{K_p(I-Q)N(\overline{\mathrm{\Omega }}\times [0,1]})$ is compact for any open bounded set $\mathrm{\Omega }\subset X$. Moreover, $QN(\overline{\mathrm{\Omega }}\times [0,1])$ is bounded. Thus, N is L-compact on $\overline{\mathrm{\Omega }}\times [0,1]$ with any open bounded set $\mathrm{\Omega }\subset X$.

In order to apply Lemma 2.1, we need to find eight appropriate open, bounded subsets ${\mathrm{\Omega }}_i$ ($i=1,2,\dots ,8$) in X.

Corresponding to the operator equation $Lu=\lambda N(u,\lambda )$, $\lambda \in (0,1)$, we have

(2.3)

(2.4)

(2.5)

Suppose that ${(u_1(t),u_2(t),u_3(t))}^T$ is a T-periodic solution of (2.3), (2.4) and (2.5) for some $\lambda \in (0,1)$.

Choose $t_i^M,t_i^m\in [0,T]$, $i=1,2,3$, such that

$$u_i\left(t_i^M\right)=\underset{t\in [0,T]}{max}{u}_i(t),u_i\left(t_i^m\right)=\underset{t\in [0,T]}{min}{u}_i(t),i=1,2,3.$$

Then it is clear that

$$u_i^{\mathrm{\prime }}\left(t_i^M\right)=0,u_i^{\mathrm{\prime }}\left(t_i^m\right)=0,i=1,2,3.$$

From this and (2.3), (2.4), (2.5), we obtain that

(2.6)

(2.7)

(2.8)

and

(2.9)

(2.10)

(2.11)

Claim A.

$$max\{u_1\left(t_1^M\right),u_2\left(t_2^M\right)\}<ln{N}_1,$$

and

$$u_3\left(t_3^M\right)<\frac{1}{{\theta }_3}ln{\left(\frac{a_3}{a_{33}}\right)}^u=ln{N}_2.$$

For $u(t_i^M)$ ($i=1,2$), there are two cases to consider.

Case 1. Assume that $u_1(t_1^M)\ge u_2(t_2^M)$, then $u_1(t_1^M)\ge u_2(t_1^M)$.

From this and (2.6), we have

$$a_1\left(t_1^M\right)-a_{11}\left(t_1^M\right)e^{{\theta }_1{u}_1(t_1^M)}>0,$$

which implies

$$e^{{\theta }_1{u}_1(t_1^M)}<\frac{a_1(t_1^M)}{a_{11}(t_1^M)}\le {\left(\frac{a_1}{a_{11}}\right)}^u.$$

That is,

$$u_2\left(t_2^M\right)\le u_1\left(t_1^M\right)<\frac{1}{{\theta }_1}ln{\left(\frac{a_1}{a_{11}}\right)}^u\le ln{N}_1.$$

Case 2. Assume that $u_1(t_1^M)<u_2(t_2^M)$, then $u_2(t_2^M)>u_1(t_2^M)$.

From this and (2.7), we have

$$a_2\left(t_2^M\right)-a_{22}\left(t_2^M\right)e^{{\theta }_2{u}_2(t_2^M)}>0,$$

which implies

$$e^{{\theta }_2{u}_2(t_2^M)}<\frac{a_2(t_2^M)}{a_{22}(t_2^M)}\le {\left(\frac{a_2}{a_{22}}\right)}^u.$$

That is,

$$u_1\left(t_1^M\right)<u_2\left(t_2^M\right)<\frac{1}{{\theta }_2}ln{\left(\frac{a_2}{a_{22}}\right)}^u\le ln{N}_1.$$

Therefore,

$$max\{u_1\left(t_1^M\right),u_2\left(t_2^M\right)\}<ln{N}_1.$$

(2.12)

For $u_3(t_3^M)$, it follows from (2.8) that

$$a_3\left(t_3^M\right)-a_{33}\left(t_3^M\right)e^{{\theta }_3{u}_3(t_3^M)}>0,$$

which implies

$$u_3\left(t_3^M\right)<\frac{1}{{\theta }_3}ln{\left(\frac{a_3}{a_{33}}\right)}^u=ln{N}_2.$$

(2.13)

Claim B.

$$u_i\left(t_i^M\right)>ln{u}_i^{+}\text{or}{u}_i\left(t_i^M\right)<ln{u}_i^{-},i=1,2,3$$

and

$$u_i\left(t_i^m\right)>ln{u}_i^{+}\text{or}{u}_i\left(t_i^m\right)<ln{u}_i^{-},i=1,2,3.$$

It follows from (2.6) that

Therefore,

From this and noticing that

$$\frac{k_1(t_1^M)}{k_1(t_1^M)+c_1(t_1^M)e^{u_1(t_1^M)}}\le \frac{k_1(t_1^M)+(1-\lambda )c_1(t_1^M)e^{u_1(t_1^M)}}{k_1(t_1^M)+c_1(t_1^M)e^{u_1(t_1^M)}}\le 1,$$

we have

which implies

$$[\frac{k_1^l}{k_1^l+c_1^u{N}_1}{\left(\frac{a_1}{k_1}\right)}^l-{\left(\frac{a_{13}}{k_1}\right)}^u{\left(\frac{a_3}{a_{33}}\right)}^u-D_1^u]-{\left(\frac{a_{11}}{k_1}\right)}^u{e}^{{\theta }_1{u}_1(t_1^M)}-\frac{H_1^u}{e^{u_1(t_1^M)}}<0.$$

From the assertion (1) of Lemma 2.3 and the above inequality, we have

$$u_1\left(t_1^M\right)>ln{u}_1^{+}\text{or}{u}_1\left(t_1^M\right)<ln{u}_1^{-}.$$

(2.14)

Similarly, from (2.9), we obtain

$$u_1\left(t_1^m\right)>ln{u}_1^{+}\text{or}{u}_1\left(t_1^m\right)<ln{u}_1^{-}.$$

(2.15)

By a similar argument, it follows from (2.7) that

$$[\frac{k_2^l}{k_2^l+c_2^u{N}_1}{\left(\frac{a_2}{k_2}\right)}^l-D_2^u]-{\left(\frac{a_{22}}{k_2}\right)}^u{e}^{{\theta }_2{u}_2(t_2^M)}-\frac{H_2^u}{e^{u_2(t_2^M)}}<0.$$

From the assertion (1) of Lemma 2.3 and the above inequality, we have

$$u_2\left(t_2^M\right)>ln{u}_2^{+}\text{or}{u}_2\left(t_2^M\right)<ln{u}_2^{-}.$$

(2.16)

Similarly, from (2.10), we obtain

$$u_2\left(t_2^m\right)>ln{u}_2^{+}\text{or}{u}_2\left(t_2^m\right)<ln{u}_2^{-}.$$

(2.17)

By a similar argument, it follows from (2.8) and (2.12) that

$$[\frac{k_3^l}{k_3^l+c_3^u{N}_2}{\left(\frac{a_3}{k_3}\right)}^l-{\left(\frac{a_{31}}{k_3}\right)}^u{N}_1^{{\theta }_1}]-{\left(\frac{a_{33}}{k_3}\right)}^u{e}^{{\theta }_3{u}_3(t_3^M)}-\frac{H_3^u}{e^{u_3(t_3^M)}}<0.$$

From the assertion (1) of Lemma 2.3 and the above inequality, we have

$$u_3\left(t_3^M\right)>ln{u}_3^{+}\text{or}{u}_3\left(t_3^M\right)<ln{u}_3^{-}.$$

(2.18)

Similarly, from (2.11), we obtain

$$u_3\left(t_3^m\right)>ln{u}_3^{+}\text{or}{u}_3\left(t_3^m\right)<ln{u}_3^{-}.$$

(2.19)

Claim C.

$$ln{l}_i^{-}<u_i\left(t_i^M\right)<ln{l}_i^{+},i=1,2,3,$$

and

$$ln{l}_i^{-}<u_i\left(t_i^m\right)<ln{l}_i^{+},i=1,2,3.$$

It follows from (2.6) that

Hence, we have

$${\left(\frac{a_1}{k_1}\right)}^u-\frac{k_1^l}{k_1^l+c_1^u{N}_1}{\left(\frac{a_{11}}{k_1}\right)}^l{e}^{{\theta }_1{u}_1(t_1^M)}-\frac{H_1^l-D_1^u{N}_1}{e^{u_1(t_1^M)}}>0.$$

From the assertion (3) of Lemma 2.3 and the above inequality, we have

$$ln{l}_1^{-}<u_1\left(t_1^M\right)<ln{l}_1^{+}.$$

(2.20)

Similarly, from (2.9), we obtain

$$ln{l}_1^{-}<u_1\left(t_1^m\right)<ln{l}_1^{+}.$$

(2.21)

By a similar argument, it follows from (2.7) that

$${\left(\frac{a_2}{k_2}\right)}^u-\frac{k_2^l}{k_2^l+c_2^u{N}_1}{\left(\frac{a_{22}}{k_2}\right)}^l{e}^{{\theta }_2{u}_2(t_2^M)}-\frac{H_2^l-D_2^u{N}_1}{e^{u_2(t_2^M)}}>0.$$

From the assertion (3) of Lemma 2.3 and the above inequality, we have

$$ln{l}_2^{-}<u_2\left(t_2^M\right)<ln{l}_2^{+}.$$

(2.22)

Similarly, from (2.10), we obtain

$$ln{l}_2^{-}<u_2\left(t_2^m\right)<ln{l}_2^{+}.$$

(2.23)

By a similar argument, it follows from (2.8) that

$${\left(\frac{a_3}{k_3}\right)}^u-\frac{k_3^l}{k_3^l+c_3^u{N}_2}{\left(\frac{a_{33}}{k_3}\right)}^l{e}^{{\theta }_3{u}_3(t_3^M)}-\frac{H_3^l}{e^{u_3(t_3^M)}}>0.$$

From the assertion (3) of Lemma 2.3 and the above inequality, we have

$$ln{l}_3^{-}<u_3\left(t_3^M\right)<ln{l}_3^{+}.$$

(2.24)

Similarly, from (2.11), we obtain

$$ln{l}_3^{-}<u_3\left(t_3^m\right)<ln{l}_3^{+}.$$

(2.25)

It follows from (2.14), (2.15), (2.20), (2.21) that

(2.26)

(2.27)

It follows from (2.16), (2.17), (2.22), (2.23) that

(2.28)

(2.29)

It follows from (2.18), (2.19), (2.24), (2.25) that

(2.30)

(2.31)

Clearly, $l_i^{\pm }$, $u_i^{\pm }$ ($i=1,2,3$) are independent of λ. Now, let us consider $QN(u,0)$ with $u={(u_1,u_2,u_3)}^T\in R^3$. Note that

$$QN(u,0)=\left(\begin{array}{c}\overline{(\frac{a_1}{k_1})}-\overline{(\frac{a_{11}}{k_1})}{e}^{{\theta }_1{u}_1}-\frac{{\overline{H}}_1}{e^{u_1}}\\ \overline{(\frac{a_2}{k_2})}-\overline{(\frac{a_{22}}{k_2})}{e}^{{\theta }_2{u}_2}-\frac{{\overline{H}}_2}{e^{u_2}}\\ \overline{(\frac{a_3}{k_3})}-\overline{(\frac{a_{33}}{k_3})}{e}^{{\theta }_3{u}_3}-\frac{{\overline{H}}_3}{e^{u_3}}\end{array}\right).$$

Letting $QN(u,0)=0$, we have

(2.32)

(2.33)

(2.34)

Therefore, it follows from the assertion (2) of Lemma 2.3 that $QN(u,0)=0$ has eight distinct solutions:

(2.35)

(2.36)

(2.37)

(2.38)

Let

Then ${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2,\dots ,{\mathrm{\Omega }}_8$ are bounded open subsets of X. It follows from (2.1) and (2.35)-(2.38) that ${\tilde{u}}_i\in {\mathrm{\Omega }}_i$ ($i=1,2,\dots ,8$). From (2.1), (2.26)-(2.31), it is easy to see that ${\overline{\mathrm{\Omega }}}_i\cap {\overline{\mathrm{\Omega }}}_j=\mathrm{\varnothing }$ ($i,j=1,2,\dots ,8$, $i\ne j$) and ${\mathrm{\Omega }}_i$ satisfies (a) in Lemma 2.1 for $i=1,2,\dots ,8$. Moreover, $QN(u,0)\ne 0$ for $u\in \partial {\mathrm{\Omega }}_i\cap KerL$. By Lemma 2.2, a direct computation gives

Here, J is taken as the identity mapping since $ImQ=KerL$. So far we have proved that ${\mathrm{\Omega }}_i$ satisfies all the assumptions in Lemma 2.1. Hence, (2.2) has at least eight T-periodic solutions ${(u_1^i(t),u_2^i(t),u_3^i(t))}^T$ ($i=1,2,\dots ,8$) and ${(u_1^i,u_2^i,u_3^i)}^T\in domL\cap {\overline{\mathrm{\Omega }}}_i$. Obviously, ${(u_1^i,u_2^i,u_3^i)}^T$ ($i=1,2,\dots ,8$) are different. Let $x_j^i(t)=e^{u_j^i(t)}$ ($j=1,2$), $y^i(t)=e^{u_3^i(t)}$ ($i=1,2,\dots ,8$). Then ${(x_1^i(t),x_2^i(t),y^i(t))}^T$ ($i=1,2,\dots ,8$) are eight different positive T-periodic solutions of (1.1). The proof is complete. □

Corollary 2.1 In addition to (H₁), (H₅), assume further that the following conditions hold:

(H₂)^∗ $\frac{k_1^l}{k_1^l+c_1^u{N}_1}{(\frac{a_1}{k_1})}^l>{(\frac{a_{13}}{k_1})}^u{(\frac{a_3}{a_{33}})}^u+D_1^u+{(\frac{a_{11}}{k_1})}^u+H_1^u$.

(H₃)^∗ $\frac{k_2^l}{k_2^l+c_2^u{N}_1}{(\frac{a_2}{k_2})}^l>D_2^u+{(\frac{a_{22}}{k_2})}^u+H_2^u$.

(H₄)^∗ $\frac{k_3^l}{k_3^l+c_3^u{N}_2}{(\frac{a_3}{k_3})}^l>{(\frac{a_{31}}{k_3})}^u{N}_1^{{\theta }_1}+{(\frac{a_{33}}{k_3})}^u+H_3^u$.

Then system (1.1) has at least eight positive T-periodic solutions.

Proof By Lemma 2.4, we have

$$(1+{\theta }_i){\left[{\left(\frac{a_{ii}}{k_i}\right)}^u\right]}^{\frac{1}{1+{\theta }_i}}{\left[\frac{H_i^u}{{\theta }_i}\right]}^{\frac{{\theta }_i}{1+{\theta }_i}}\le {\left(\frac{a_{ii}}{k_i}\right)}^u+H_i^u,i=1,2,3.$$

Therefore, the conditions in Theorem 2.1 are satisfied. □

Example 2.2 In (1.1), take

Then we have

Therefore,

Hence, the conditions in Corollary 2.1 are satisfied. By Corollary 2.1, system (1.1) has at least eight positive four-periodic solutions.