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Existence and multiplicity of positive periodic solutions of ratio-dependent food chain model

Boundary Value Problems20162016:144

https://doi.org/10.1186/s13661-016-0653-0

  • Received: 18 November 2015
  • Accepted: 28 July 2016
  • Published:

Abstract

By utilizing the coincidence degree theory and the related continuation theorem, as well as some prior estimates, we investigate the existence and multiplicity of positive periodic solutions of ratio-dependent food chain model with exploited terms. Some sufficient criteria are established for the existence and multiplicity of periodic solutions.

Keywords

  • coincidence degree
  • periodic solution
  • exploited term
  • ratio-dependent

MSC

  • 34K13
  • 92D25

1 Introduction

The last years have seen very important progress made on Michaelis-Menten type ratio-dependent predator-prey model in mathematical ecology literature, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance and usually takes the form
$$ x^{\prime}(t)=x (r-kx )-c\frac {mxy}{ay+x},\qquad y^{\prime}(t) = \biggl(\frac{mx}{ay+x}-d \biggr)y, $$
(1.1)
where x, y stand for prey and predator density, respectively, r, k, a, c, d, m are positive constants that stand for prey intrinsic growth rate, carrying capacity, half-saturation constant, conversion rate, predator’s death rate, and maximal predator growth rate, respectively. System (1.1) is capable of producing far richer and biologically more realistic dynamics. Specifically, it will not produce the paradox of biological control and the paradox of enrichment. In view of these features it has been studied by many authors leading to great progress [17]. Moreover, the ratio-dependence form is applied successfully to some other models, for example, in [8], the authors investigated the following three trophic level food chain model with ratio dependence:
$$\begin{aligned}& x^{\prime}(t)=x \biggl(r-kx-\frac{b_{1} y}{m_{1}y+x} \biggr), \\& y^{\prime}(t)=y \biggl(\frac{c_{1}x}{m_{1}y+x}-d_{1}- \frac{b_{2}z}{ m_{2}z+y} \biggr), \\& z^{\prime}(t)=z \biggl(\frac{c_{2}y}{m_{2}z+y}-d_{2} \biggr), \end{aligned}$$
(1.2)
where x, y, z stand for the population densities of prey, predator and top predator, respectively. For \(i=1,2\), \(m_{i}\), \(d_{i}\), \(b_{i}\), \(c_{i}\) are half-saturation constants and the death rates of predator, capture rates, and maximal predator growth rates, respectively, \(r/k\) gives the carrying capacity of the prey. This model reflects the simple relation of these three species: z prey on y and only on y, and y prey on x and nutrient recycling is not accounted for. It was shown that this model is rich in boundary dynamics and is capable of generating extinction dynamics.
Recently, there has been a rich body of literature on ecological systems with exploited term(s) and numerous good results have been obtained, for example; see [6, 7, 913]. In these references, instead of studying the existence of a periodic solution, one investigated the existence of multiple periodic solutions for considering the inclusion of the effect of periodic changing environment. This is due to the fact that it is more likely for some biological species to take on multiple periodic change regulations and have multiple local stable periodic phenomena. In the present paper, we study the following ratio-dependent food chain model with exploited terms in a periodically varying environment because the variation of the environment plays an important role in many biological and ecological systems:
$$\begin{aligned}& x^{\prime}(t)=x(t) \biggl[ r(t)-k(t)x(t)-\frac {b_{1}(t)y(t)}{m_{1}(t)y(t)+x(t)} \biggr] -h_{1}(t), \\& y^{\prime}(t)=y(t) \biggl[ \frac {c_{1}(t)x(t)}{m_{1}(t)y(t)+x(t)}-d_{1}(t)- \frac{b_{2}(t)z(t)}{ m_{2}(t)z(t)+y(t)} \biggr] -h_{2}(t), \\& z^{\prime}(t)=z(t) \biggl[ \frac {c_{2}(t)y(t)}{m_{2}(t)z(t)+y(t)}-d_{2}(t) \biggr] -h_{3}(t), \end{aligned}$$
(1.3)
where \(h_{1}\), \(h_{2}\), \(h_{3}\) are nonnegative continuous ω-periodic functions representing exploited terms, the other variables and parameters have the same biological meanings as in system (1.2) except that these parameters are ω-periodic functions now.

The paper is organized as follows. In Section 2, the original contributions of this work are summarized. In Section 3, some conclusions are given. Finally, the proofs of our main results are reported in the Appendix to close this paper.

2 Main results

We are now ready to present the main contributions involving eight theorems. For simplicity, we will discuss in detail for Theorem 2.1, the remainder results are similar and their proofs are presented in the Appendix.

For the reader’s convenience, we now recall Mawhin’s coincidence degree [14], which our study is based upon.

Let X, Z be normed vector spaces, \(L: \operatorname{Dom}L\subset X\rightarrow Z\) be a linear mapping, \(N: X\rightarrow Z\) be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if \(\operatorname{dim} \operatorname{Ker}L=\operatorname{codim} \operatorname{Im}L<+\infty\) and ImL is closed in Z. If L is a Fredholm mapping of index zero there exist continuous projectors \(P: X\rightarrow X\) and \(Q: Z\rightarrow Z\) such that \(\operatorname{Im}P=\operatorname{Ker}L\), \(\operatorname{Im}L=\operatorname{Ker}Q=\operatorname{Im}(I-Q)\). It follows that \(L|\operatorname{Dom}L\cap \operatorname{Ker}P: (I-P)X\rightarrow \operatorname{Im}L\) 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 Ω̄ if \(QN(\bar{\Omega})\) is bounded and \(K_{P}(I-Q)N: \bar{\Omega }\rightarrow X\) is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism \(J: \operatorname{Im}Q\rightarrow \operatorname{Ker}L\).

Theorem A

(Continuation theorem)

Let L be a Fredholm mapping of index zero and let N be L-compact on Ω̄. Suppose:
  1. (i)

    For each \(\lambda\in(0,1)\), every solution x of \(Lx=\lambda Nx\) is such that \(x\notin \partial\Omega\);

     
  2. (ii)

    \(QNx\neq0\) for each \(x\in\partial\Omega\cap \operatorname{Ker}L\);

     
  3. (iii)

    \(\deg\{JQN, \Omega\cap \operatorname{Ker}L, 0\}\neq0\).

     
Then the equation \(Lx=Nx\) has at least one solution lying in \(\operatorname{Dom}L\cap\bar{\Omega}\).
For a bounded continuous function \(g(t)\) on \(\mathbb{R}\), we use the following notations:
$$ g^{U}=\max_{t\in[0,\omega]}g(t),\qquad g^{\ell}=\min _{t\in[0,\omega]}g(t), $$
where \(g(t)\) is s continuous function.

Theorem 2.1

If \(h_{1}(t)\neq0\), \(h_{2}(t)\neq0\), \(h_{3}(t)\neq0\), and the following conditions are satisfied:
$$\begin{aligned} (\mathrm{H}1)&\quad r^{\ell}- \biggl(\frac{b_{1}}{m_{1}} \biggr)^{U}>2\sqrt{k^{U} h_{1}^{U}} , \\ (\mathrm{H}2)&\quad \biggl[ c_{1}^{\ell}-d_{1}^{U}- \biggl(\frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] \frac{h_{1}^{\ell}}{r^{U}}-m_{1}^{U}h_{2}^{U}>2 \sqrt {m_{1}^{U} \biggl[ d_{1}^{U}+ \biggl(\frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] \frac{r^{U} h_{2}^{U} }{k^{\ell}}} , \\ (\mathrm{H}3) &\quad \bigl(c_{2}^{\ell}-d_{2}^{U} \bigr) \frac{h_{2}^{\ell}}{c_{1}^{U}}-m_{2}^{U} h_{3}^{U}>2 \sqrt{\frac{r^{U} c_{1}^{U} m_{2}^{U} d_{2}^{U} h_{3}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}}} . \end{aligned}$$
Then system (1.3) has at least eight positive periodic solutions.

Proof

We make the change of variables
$$x(t)=\exp\bigl\{ u(t)\bigr\} , \qquad y(t)=\exp\bigl\{ v(t)\bigr\} ,\qquad z(t)=\exp \bigl\{ w(t)\bigr\} . $$
Then system (1.3) can be written as
$$ \textstyle\begin{cases} u^{\prime}(t)=r(t)-k(t)e^{u(t)}-\frac {b_{1}(t)e^{v(t)}}{m_{1}(t)e^{v(t)}+e^{u(t)}}-\frac{h_{1}(t)}{e^{u(t)}}, \\ v^{\prime}(t)=\frac{c_{1}(t)e^{u(t)}}{m_{1}(t)e^{v(t)}+e^{u(t)}}-d_{1}(t)-\frac {b_{2}(t)e^{w(t)}}{m_{2}(t)e^{w(t)}+e^{v(t)}}- \frac{h_{2}(t)}{e^{v(t)}}, \\ w^{\prime}(t)=\frac{c_{2}(t)e^{v(t)}}{m_{2}(t)e^{w(t)}+e^{v(t)}}-d_{2}(t)-\frac {h_{3}(t)}{e^{w(t)}}. \end{cases} $$
(2.1)
It is easy to see that if system (2.1) has an ω-periodic solution \((u^{*},v^{*},w^{*})^{T}\), then \((x^{*},y^{*}, z^{*})^{T}=(e^{u^{*}},e^{v^{*}},e^{w^{*}})^{T}\) is a positive ω-periodic solution of system (1.3). To this end, it suffices to prove that system (2.1) has at least eight ω-periodic solutions.
For \(\lambda\in(0,1)\), we consider the following system:
$$ \textstyle\begin{cases} u^{\prime}(t)=\lambda [ r(t)-k(t)e^{u(t)}-\frac{b_{1}(t)e^{v(t)}}{m_{1}(t)e^{v(t)}+e^{u(t)}}-\frac {h_{1}(t)}{e^{u(t)}} ] , \\ v^{\prime}(t)=\lambda [ \frac {c_{1}(t)e^{u(t)}}{m_{1}(t)e^{v(t)}+e^{u(t)}}-d_{1}(t)-\frac {b_{2}(t)e^{w(t)}}{m_{2}(t)e^{w(t)}+e^{v(t)}}- \frac{h_{2}(t)}{e^{v(t)}} ] , \\ w^{\prime}(t)=\lambda [ \frac {c_{2}(t)e^{v(t)}}{m_{2}(t)e^{w(t)}+e^{v(t)}}-d_{2}(t)-\frac {h_{3}(t)}{e^{w(t)}} ] . \end{cases} $$
(2.2)
Suppose that \((u(t),v(t),w(t))^{T}\) is an arbitrary ω-periodic solution of system (2.2) for a certain \(\lambda\in(0,1)\). Then we can choose \(\xi_{i}\), \(\eta_{i}\), \(i=1, 2, 3\) such that
$$\begin{aligned}& u(\xi_{1})=\max_{t\in [0,\omega]}\bigl\{ u(t)\bigr\} , \qquad u(\eta_{1})=\min_{t\in [0,\omega]}\bigl\{ u(t)\bigr\} , \end{aligned}$$
(2.3)
$$\begin{aligned}& v(\xi_{2})=\max_{t\in[0,\omega]}\bigl\{ v(t)\bigr\} ,\qquad v(\eta_{2})=\min_{t\in[0,\omega]}\bigl\{ v(t)\bigr\} , \end{aligned}$$
(2.4)
$$\begin{aligned}& w(\xi_{3})=\max_{t\in[0,\omega]}\bigl\{ w(t)\bigr\} ,\qquad w(\eta_{3})=\min_{t\in[0,\omega]}\bigl\{ w(t)\bigr\} . \end{aligned}$$
(2.5)
By the first equation of (2.2) and (2.3), we have
$$ r^{U}\geq r(\xi_{1})>k(\xi_{1})e^{u(\xi_{1})} \geq k^{\ell}e^{u(\xi_{1})}, $$
which reduces to
$$ u(\xi_{1})< \ln \biggl\{ \frac{r^{U}}{k^{\ell}} \biggr\} . $$
(2.6)
Again from the first equation of (2.2) and (2.3), it follows that
$$ h_{1}^{\ell}e^{-u(\eta_{1})}\leq h_{1}( \eta_{1})e^{-u(\eta_{1})}< r(\eta_{1})\leq r^{U}, $$
which implies
$$ u(\eta_{1})>\ln \biggl\{ \frac{ h_{1}^{\ell}}{r^{U}} \biggr\} . $$
(2.7)
From the second equation of (2.2) and (2.4), (2.6), we obtain
$$ d_{1}^{\ell}\leq d_{1}(\xi_{2})< \frac{c_{1}(\xi_{2}) e^{u(\xi_{2})}}{m_{1}(\xi_{2})e^{v(\xi_{2})}}< \frac{r^{U} c_{1}^{U}}{k^{\ell}m_{1}^{\ell}e^{v(\xi_{2})}}, $$
which reduces to
$$ v(\xi_{2})< \ln \biggl\{ \frac{r^{U} c_{1}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \biggr\} . $$
(2.8)
Moreover, from the second equation of (2.2) and (2.4), we get
$$ h_{2}^{\ell}e^{-v(\eta_{2})}\leq h_{2}( \eta_{2}) e^{-v(\eta_{2})}< c_{1}(\eta_{2})\leq c_{1}^{U}, $$
that is,
$$ v(\eta_{2})>\ln \biggl\{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \biggr\} . $$
(2.9)
From the third equation of (2.2), (2.5), and (2.8), we have
$$ d_{2}^{\ell}\leq d_{2}(\xi_{3})< \frac{c_{2}(\xi_{3})e^{v(\xi_{3})}}{m_{2}(\xi_{3})e^{w(\xi _{3})}}< \frac {r^{U}c_{1}^{U}c_{2}^{U}}{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}e^{w(\xi_{3})}}, $$
which implies
$$ w(\xi_{3})< \ln \biggl\{ \frac{r^{U}c_{1}^{U}c_{2}^{U}}{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \biggr\} . $$
(2.10)
It follows from the third equation of (2.2) and (2.5) that
$$ h_{3}^{\ell}e^{-w(\eta_{3})}\leq h_{3}( \eta_{3}) e^{-w(\eta_{3})}< c_{2}(\eta_{3})\leq c_{2}^{U}, $$
which reduces to
$$ w(\eta_{3})>\ln \biggl\{ \frac{h_{3}^{\ell}}{c_{2}^{U}} \biggr\} . $$
(2.11)
Furthermore, by the definition of \(\xi_{1}\) and the first equation of (2.2), we know
$$ r(\xi_{1})-k(\xi_{1})e^{u(\xi_{1})}-\frac{b_{1}(\xi_{1})e^{v(\xi _{1})}}{m_{1}(\xi _{1})e^{v(\xi_{1})} +e^{u(\xi_{1})}}- \frac{h_{1}(\xi_{1})}{e^{u(\xi_{1})}}=0. $$
Then
$$ k^{U} e^{2u(\xi_{1})}+ \biggl[ \biggl(\frac{b_{1}}{m_{1}} \biggr)^{U}-r^{\ell}\biggr] e^{u(\xi_{1})}+h_{1}^{U}>0, $$
which produces
$$ u(\xi_{1})>\ln A_{0}^{+} \quad \mbox{or}\quad u(\xi_{1})< \ln A_{0}^{-}, $$
(2.12)
where
$$ A_{0}^{\pm}=\frac{1}{2k^{U}} \biggl\{ \biggl[ r^{\ell}- \biggl(\frac{b_{1}}{m_{1}} \biggr)^{U} \biggr] \pm \sqrt{ \biggl[ r^{\ell}- \biggl(\frac{b_{1}}{m_{1}} \biggr)^{U} \biggr] ^{2} -4k^{U} h_{1}^{U}} \biggr\} . $$
By the definition of \(\eta_{1}\) and the parallel argument to (2.12), it is easy to prove that
$$ u(\eta_{1})>\ln A_{0}^{+} \quad \mbox{or}\quad u(\eta_{1})< \ln A_{0}^{-}. $$
(2.13)
Similarly, by the definition of \(\xi_{2}\) and the second equation of (2.2), we have
$$ d_{1}(\xi_{2})+\frac{b_{2}(\xi_{2})e^{w(\xi_{2})}}{m_{2}(\xi_{2})e^{w(\xi _{2})}+e^{v(\xi_{2})}}+ h_{2}( \xi_{2})e^{-v(\xi_{2})}-\frac{c_{1}(\xi_{2})e^{u(\xi_{2})}}{m_{1}(\xi _{2})e^{v(\xi_{2})}+e^{u(\xi_{2})}}=0. $$
Then
$$ m_{1}^{U} \biggl[ d_{1}^{U}+ \biggl( \frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] e^{2v(\xi_{2})}- \biggl\{ \biggl[ c_{1}^{\ell}-d_{1}^{U}- \biggl( \frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] \frac {h_{1}^{\ell}}{r^{U}}-h_{2}^{U}m_{1}^{U} \biggr\} e^{v(\xi_{2})}+\frac{r^{U} h_{2}^{U}}{k^{\ell}}>0. $$
Solving the inequality, we get
$$ v(\xi_{2})>\ln B_{0}^{+} \quad \mbox{or}\quad v(\xi_{2})< \ln B_{0}^{-}, $$
(2.14)
where
$$\begin{aligned} B_{0}^{\pm} =&\frac{1}{2m_{1}^{U} [ d_{1}^{U}+ (\frac{b_{2}}{m_{2}} )^{U} ] }\biggl\{ \biggl[ c_{1}^{\ell}-d_{1}^{U}- \biggl( \frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] \frac{h_{1}^{\ell}}{r^{U}}-m_{1}^{U} h_{2}^{U} \\ &{}\pm\sqrt{ \biggl\{ \biggl[ c_{1}^{\ell}-d_{1}^{U}- \biggl(\frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] \frac{h_{1}^{\ell}}{r^{U}}-m_{1}^{U}h_{2}^{U} \biggr\} ^{2}-4m_{1}^{U} \biggl[ d_{1}^{U}+ \biggl(\frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] \frac{h_{2}^{U} r^{U}}{k^{\ell}}}\biggr\} . \end{aligned}$$
In the same way, we can obtain
$$ v(\eta_{2})>\ln B_{0}^{+} \quad \mbox{or}\quad v(\eta_{2})< \ln B_{0}^{-}. $$
(2.15)
Using the definition of \(\xi_{3}\) and the third equation of (2.2), we get
$$ d_{2}(\xi_{3})m_{2}(\xi_{2})e^{2w(\xi_{3})}+ \bigl[d_{2}(\xi_{3})-c_{2}(\xi_{3}) \bigr]e^{v(\xi _{3})+w(\xi_{3})}+h_{3}(\xi_{3})e^{v(\xi_{3})} +m_{2}(\xi_{3})h_{3}(\xi_{3})e^{w(\xi_{3})}=0, $$
which, combined with (2.8) and (2.9), yields
$$ d_{2}^{U} m_{2}^{U}e^{2w(\xi_{3})}+h_{3}^{U} \frac{r^{U} c_{1}^{U} }{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}}- \biggl[ \bigl(c_{2}^{\ell}-d_{2}^{U} \bigr)\frac{h_{2}^{\ell}}{c_{1}^{U}}-m_{2}^{U} h_{3}^{U} \biggr] e^{w(\xi_{3})}>0. $$
Solving the inequality, we have
$$ w(\xi_{3})>\ln C_{0}^{+} \quad \mbox{or} \quad w(\xi_{3})< \ln C_{0}^{-}, $$
(2.16)
where
$$\begin{aligned} C_{0}^{\pm} =&\frac{1}{2d_{2}^{U}m_{2}^{U}} \biggl\{ \bigl(c_{2}^{\ell}-d_{2}^{U}\bigr) \frac{h_{2}^{\ell}}{c_{1}^{U}}-m_{2}^{U} h_{3}^{U} \\ &{}\pm \sqrt{ \biggl[ \bigl(c_{2}^{\ell}-d_{2}^{U} \bigr) \frac{h_{2}^{\ell}}{c_{1}^{U}}-m_{2}^{U} h_{3}^{U} \biggr] ^{2}-4\frac{r^{U} c_{1}^{U} m_{2}^{U} d_{2}^{U} h_{3}^{U} }{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}}} \biggr\} . \end{aligned}$$
Likewise, it follows that
$$ w(\eta_{3})>\ln C_{0}^{+}\quad \mbox{or}\quad w(\eta_{3})< \ln C_{0}^{-}. $$
(2.17)
From (2.6), (2.7), (2.12), and (2.13), we obtain, for any \(t\in[0,\omega]\),
$$ \ln \biggl\{ \frac{h_{1}^{\ell}}{r^{U}} \biggr\} < u(t)< \ln A_{0}^{-} $$
(2.18)
or
$$ \ln A_{0}^{+}< u(t)< \ln \biggl\{ \frac{r^{U}}{k^{\ell}} \biggr\} . $$
(2.19)
From (2.8), (2.9), (2.14), and (2.15), we obtain, for any \(t\in[0,\omega]\),
$$ \ln \biggl\{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \biggr\} < v(t)< \ln B_{0}^{-} $$
(2.20)
or
$$ \ln B_{0}^{+}< v(t)< \ln \biggl\{ \frac{r^{U} c_{1}^{U} }{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \biggr\} . $$
(2.21)
From (2.10), (2.11), (2.16), and (2.17), we obtain, for any \(t\in[0,\omega]\),
$$ \ln \biggl\{ \frac{h_{3}^{U}}{c_{2}^{U}} \biggr\} < w(t)< \ln C_{0}^{-} $$
(2.22)
or
$$ \ln C_{0}^{+}< w(t)< \ln \biggl\{ \frac{r^{U} c_{1}^{U} c_{2}^{U} }{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \biggr\} . $$
(2.23)
It is easily seen that \(\ln A_{0}^{\pm}\), \(\ln B_{0}^{\pm}\), \(\ln C_{0}^{\pm}\), \(\ln \{ \frac{h_{1}^{\ell}}{r^{U}} \} \), \(\ln \{ \frac{r^{U}}{k^{\ell}} \}\), \(\ln \{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \}\), \(\ln \{ \frac{r^{U} c_{1}^{U} }{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \}\), \(\ln \{ \frac{h_{3}^{U}}{c_{2}^{U}} \}\), \(\ln \{ \frac{r^{U} c_{1}^{U}c_{2}^{U} }{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \} \) are independent of λ.

In the following, we will show that (i)-(iii) in Theorem A are satisfied.

First, let us take
$$ X=Z=\bigl\{ \bigl(u(t),v(t),w(t)\bigr)^{T}\in C\bigl( \mathbb{R}, \mathbb{R}^{3}\bigr)|u(t+\omega)=u(t),v(t+\omega )=v(t),w(t+\omega )=w(t)\bigr\} $$
and
$$ \bigl\Vert \bigl(u(t),v(t),w(t)\bigr)^{T}\bigr\Vert =\max _{t\in[0,\omega]}\bigl\vert u(t)\bigr\vert +\max_{t\in [0,\omega]} \bigl\vert v(t)\bigr\vert +\max_{t\in[0,\omega]}\bigl\vert w(t)\bigr\vert . $$
Then X and Z are Banach spaces equipped with the norm \(\|\cdot\|\).
Let
$$ L\left ( \textstyle\begin{array}{@{}c@{}} u(t)\\ v(t)\\ w(t) \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{@{}c@{}} u^{\prime}(t) \\ v^{\prime}(t)\\ w^{\prime}(t) \end{array}\displaystyle \right ), $$
where \(\operatorname{Dom}L=\{ (u,v,w)^{T}\in X:(u,v,w)^{T}\in C^{1}(\mathbb{R},\mathbb{R}^{3})\}\).
Define \(N:X\rightarrow X\) as follows:
$$N\left ( \textstyle\begin{array}{@{}c@{}} u(t)\\ v(t)\\ w(t) \end{array}\displaystyle \right )= \left ( \textstyle\begin{array}{@{}c@{}} r(t)-k(t)e^{u(t)}-\frac{b_{1}(t)e^{v(t)}}{m_{1}(t)e^{v(t)} +e^{u(t)}}-\frac{h(t)}{e^{u(t)}} \\ \frac{c_{1}(t)e^{u(t)}}{m_{1}(t)e^{v(t)}+e^{u(t)}}-d_{1}(t)-\frac{b_{2}(t)e^{w(t)}}{m_{2}(t)e^{w(t)}+e^{v(t)}}-\frac{h_{2}(t)}{e^{v(t)}} \\ \frac{c_{2}(t)e^{v(t)}}{m_{2}(t)e^{w(t)}+e^{v(t)}}-d_{2}(t)-\frac{h_{3}(t)}{e^{w(t)}} \end{array}\displaystyle \right ):=\left ( \textstyle\begin{array}{@{}c@{}} N_{1}(t)\\ N_{2}(t)\\ N_{3}(t) \end{array}\displaystyle \right ). $$
Define projectors P and Q by
$$P\left ( \textstyle\begin{array}{@{}c@{}} u(t)\\ v(t)\\ w(t) \end{array}\displaystyle \right )=Q\left ( \textstyle\begin{array}{@{}c@{}} u(t)\\ v(t)\\ w(t) \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{@{}c@{}}\frac{1}{\omega}\int_{0}^{\omega}u(t)\,dt\\ \frac{1}{\omega}\int_{0}^{\omega}v(t)\,dt\\ \frac{1}{\omega}\int _{0}^{\omega}w(t)\,dt \end{array}\displaystyle \right ), \quad \left ( \textstyle\begin{array}{@{}c@{}} u(t)\\ v(t)\\ w(t) \end{array}\displaystyle \right ) \in X. $$
Then it follows that \(\operatorname{Ker}L=\mathbb{R}^{3}\), \(\operatorname{Im} L=\operatorname{Ker} Q=\{(u(t),v(t))^{T}\in X :\bar{u}=\bar{v}=\bar{w}=0\}\) is closed in X, and \(\dim \operatorname{Ker} L=3=\operatorname{codim} \operatorname{Im} L\), and P, Q are continuous projectors such that
$$\operatorname{Im}P=\operatorname{Ker} L, \qquad \operatorname{Ker}Q= \operatorname{Im}L=\operatorname{Im}(I-Q). $$
Hence, L is a Fredholm operator of index zero. Furthermore, the generalized inverse (to L) \(K_{P}: \operatorname{Im} L\rightarrow \operatorname{Dom} L\cap \operatorname{Ker} P\) is given by
$$K_{P}\left ( \textstyle\begin{array}{@{}c@{}} u(t)\\ v(t)\\ w(t) \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{@{}c@{}}\int_{0}^{t} u(s)\,ds-\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t} u(s)\,ds\,dt\\ \int_{0}^{t} v(s)\,ds-\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t} v(s)\,ds\,dt\\ \int_{0}^{t} w(s)\,ds-\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t} w(s)\,ds\,dt \end{array}\displaystyle \right ). $$
Then
$$QN\left ( \textstyle\begin{array}{@{}c@{}}u(t)\\ v(t)\\ w(t) \end{array}\displaystyle \right )= \left ( \textstyle\begin{array}{@{}c@{}}\frac{1}{\omega}\int_{0}^{\omega}N_{1}(s)\,ds\\ \frac{1}{\omega}\int_{0}^{\omega}N_{2}(s)\,ds \\ \frac{1}{\omega}\int_{o}^{\omega}N_{3}(s)\,ds \end{array}\displaystyle \right ) $$
and
$$K_{p}(I-Q)N\left ( \textstyle\begin{array}{@{}c@{}}u(t)\\ v(t)\\ w(t) \end{array}\displaystyle \right )= \left ( \textstyle\begin{array}{@{}c@{}}\int_{0}^{\omega}N_{1}(t)\,dt-\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t} N_{1}(s)\,ds\,dt+(\frac{1}{2}-\frac{t}{\omega})\int_{0}^{\omega}N_{1}(s)\,ds\\ \int _{0}^{\omega}N_{2}(t)\,dt-\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t} N_{2}(s)\,ds\,dt+(\frac{1}{2}-\frac{t}{\omega})\int_{0}^{\omega}N_{2}(s)\,ds\\ \int _{0}^{\omega}N_{3}(t)\,dt-\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t} N_{3}(s)\,ds\,dt+(\frac{1}{2}-\frac{t}{\omega})\int_{0}^{\omega}N_{3}(s)\,ds \end{array}\displaystyle \right ). $$
Now, we reach the point where we search for appropriate open bounded subsets \(\Omega_{i}\), \(i=1,2,\ldots,8\), for the application of the continuation theorem. To this end, we take
$$\begin{aligned}& \Omega_{1}=\left \{ (u,v,w)^{T}\in X\left \vert \textstyle\begin{array}{l} u(t)\in (\ln \{ \frac{h_{1}^{\ell}}{r^{U}} \} ,\ln A_{0}^{-} )\\ v(t)\in (\ln \{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \} ,\ln B_{0}^{-} )\\ w(t)\in (\ln \{ \frac{h_{3}^{U}}{c_{2}^{U}} \} ,\ln C_{0}^{-} ) \end{array}\displaystyle \right . \right \} , \\& \Omega_{2}=\left \{ (u,v,w)^{T}\in X\left \vert \textstyle\begin{array}{l} u(t)\in (\ln \{ \frac{h_{1}^{\ell}}{r^{U}} \} ,\ln A_{0}^{-} )\\ v(t)\in (\ln \{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \} ,\ln B_{0}^{-} )\\ w(t)\in (\ln C_{0}^{+},\ln \{ \frac{r^{U} c_{1}^{U} c_{2}^{U}}{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \} ) \end{array}\displaystyle \right . \right \} , \\& \Omega_{3}=\left \{ (u,v,w)^{T}\in X\left \vert \textstyle\begin{array}{l} u(t)\in (\ln \{ \frac{h_{1}^{\ell}}{r^{U}} \} ,\ln A_{0}^{-} )\\ v(t)\in (\ln B_{0}^{+},\ln \{ \frac{r^{U} c_{1}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \} )\\ w(t)\in (\ln \{ \frac{h_{3}^{U}}{c_{2}^{U}} \} ,\ln C_{0}^{-} ) \end{array}\displaystyle \right . \right \} , \\& \Omega_{4}=\left \{ (u,v,w)^{T}\in X\left \vert \textstyle\begin{array}{l} u(t)\in (\ln \{ \frac{h_{1}^{\ell}}{r^{U}} \} ,\ln A_{0}^{-} )\\ v(t)\in (\ln B_{0}^{+},\ln \{ \frac{r^{U} c_{1}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \} )\\ w(t)\in (\ln C_{0}^{+},\ln \{ \frac{r^{U} c_{1}^{U} c_{2}^{U}}{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \} ) \end{array}\displaystyle \right . \right \} , \\& \Omega_{5}=\left \{ (u,v,w)^{T}\in X\left \vert \textstyle\begin{array}{l} u(t)\in (\ln A_{0}^{+},\ln \{ \frac{r^{U}}{k^{\ell}} \} )\\ v(t)\in (\ln \{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \} ,\ln B_{0}^{-} )\\ w(t)\in (\ln \{ \frac{h_{3}^{U}}{c_{2}^{U}} \} ,\ln C_{0}^{-} ) \end{array}\displaystyle \right . \right \} , \\& \Omega_{6}=\left \{ (u,v,w)^{T}\in X\left \vert \textstyle\begin{array}{l} u(t)\in (\ln A_{0}^{+},\ln \{ \frac{r^{U}}{k^{\ell}} \} )\\ v(t)\in (\ln \{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \} ,\ln B_{0}^{-} )\\ w(t)\in (\ln C_{0}^{+},\ln \{ \frac{r^{U} c_{1}^{U} c_{2}^{U}}{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \} ) \end{array}\displaystyle \right . \right \} , \\& \Omega_{7}=\left \{ (u,v,w)^{T}\in X\left \vert \textstyle\begin{array}{l} u(t)\in (\ln A_{0}^{+},\ln \{ \frac{r^{U}}{k^{\ell}} \} )\\ v(t)\in (\ln B_{0}^{+},\ln \{ \frac{r^{U} c_{1}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \} )\\ w(t)\in (\ln \{ \frac{h_{3}^{U}}{c_{2}^{U}} \} ,\ln C_{0}^{-} ) \end{array}\displaystyle \right . \right \} , \\& \Omega_{8}=\left \{ (u,v,w)^{T}\in X\left \vert \textstyle\begin{array}{l} u(t)\in (\ln A_{0}^{+},\ln \{ \frac{r^{U}}{k^{\ell}} \} )\\ v(t)\in (\ln B_{0}^{+},\ln \{ \frac{r^{U} c_{1}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \} )\\ w(t)\in (\ln C_{0}^{+},\ln \{ \frac{r^{U} c_{1}^{U} c_{2}^{U}}{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \} ) \end{array}\displaystyle \right . \right \} . \end{aligned}$$
Then \(\Omega_{i}\) (\(i=1,\ldots,8\)) are bounded open subset of X, \(\Omega_{i}\cap\Omega_{j}=\phi\), \(i\neq j\), \(i,j=1,\ldots,8\). Hence \(\Omega_{i}\) (\(i=1,\ldots,8\)) satisfies the requirement (i) in Theorem A.
Second, we will prove that (ii) holds. If it is not true, then when \((u,v,w)^{T} \in\partial\Omega_{i}\cap \operatorname{Ker} L=\partial\Omega_{i}\cap \mathbb{R}^{3}\), \(i=1,\ldots,8\), \(QNx\neq0\). There exist three points \(t_{1},t_{2},t_{3}\in[0,\omega]\) such that
$$ \textstyle\begin{cases} r(t_{1})-k(t_{1})e^{u}-\frac{b_{1}(t_{1})e^{v}}{m_{1}(t_{1})e^{v}+e^{u}}-\frac {h_{1}(t_{1})}{e^{u}}=0, \\ \frac{c_{1}(t_{2})e^{u}}{m_{1}(t_{2})e^{v}+e^{u}}-d_{1}(t_{2})- \frac{b_{2}(t_{2})e^{w}}{m_{2}(t_{2})e^{w}+e^{v}}- \frac{h_{2}(t_{2})}{e^{v}}=0, \\ \frac{c_{2}(t_{3})e^{v}}{m_{2}(t_{3})e^{w}+e^{v}}-d_{2}(t_{3})-\frac{h_{3}(t_{3})}{e^{w}}=0. \end{cases} $$
From the above arguments, we have
$$\begin{aligned}& \ln \biggl\{ \frac{h_{1}^{\ell}}{r^{U}} \biggr\} < u(t)< \ln A_{0}^{-} \quad \mbox{or}\quad \ln A_{0}^{+}< u(t)< \ln \biggl\{ \frac{r^{U}}{k^{\ell}} \biggr\} , \\& \ln \biggl\{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \biggr\} < v(t)< \ln B_{0}^{-} \quad \mbox{or} \quad \ln B_{0}^{+}< v(t)< \ln \biggl\{ \frac{r^{U} c_{1}^{U} }{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \biggr\} , \\& \ln \biggl\{ \frac{h_{3}^{U}}{c_{2}^{U}} \biggr\} < w(t)< \ln C_{0}^{-} \quad \mbox{or} \quad \ln C_{0}^{+}< w(t)< \ln \biggl\{ \frac{r^{U} c_{1}^{U} c_{2}^{U} }{ k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \biggr\} . \end{aligned}$$
Then we know \((u,v,w)^{T}\) belongs to one of \(\Omega_{i}\cap\mathbb{R}^{3}\), \(i=1,\ldots,8\). This leads to a contradiction.

Finally, we show that (iii) in Theorem A is satisfied. We proceed in our proofs by two steps.

On one hand, we show that, for \(i=1,\ldots,8\),
$$\begin{aligned} \deg\bigl\{ JQNx,\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} =& \deg\bigl\{ \bigl(N_{1}(t_{1}),N_{2}(t_{2}),N_{3}(t_{3}) \bigr)^{T},\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} \\ =& \deg\bigl\{ (\widehat{N}_{1},\widehat{N}_{2},\widehat {N}_{3})^{T},\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} . \end{aligned}$$
(2.24)
Here
$$\left [ \textstyle\begin{array}{@{}c@{}} \widehat{N}_{1}\\ \widehat{N}_{2}\\ \widehat{N}_{3} \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{@{}c@{}} \hat{r}-\hat{k}e^{u}-\hat{h}_{1} e^{-u}\\ (\hat{c}_{1}-\hat{d}_{1} )e^{u+v}-\widehat {m}_{1}\hat{d}_{1} e^{2v}-\hat{h}_{2} e^{u}-\widehat{m}_{1}\hat{h}_{2} e^{v}\\ (\hat{c}_{2}-\hat{d}_{2}) e^{v+w}-\hat{h}_{3}(\widehat{m}_{2} e^{w}+e^{v})-\widehat{m}_{2}\hat{d}_{2} e^{2w} \end{array}\displaystyle \right ] , $$
and , , \(\hat{b}_{i}\), \(\hat{c}_{i}\), \(\widehat {m}_{i}\) (\(i=1,2\)), \(\hat{h}_{j}\), \(j=1,2,3\) are some chosen positive constants satisfying the following conditions:
$$\begin{aligned}& \hat{r}k^{\ell}< r^{U} \hat{k},\qquad \hat{r}h_{1}^{\ell}< r^{U} \hat{h}_{1},\qquad \hat{c}_{1}m_{1}^{\ell}d_{1}^{\ell}< c_{1}^{U}\widehat{m}_{1} \hat{d}_{1}, \\& \hat{c}_{1} h_{2}^{\ell}< c_{1}^{U} \hat{h}_{2},\qquad \hat{c}_{2}m_{2}^{\ell}d_{2}^{\ell}< c_{2}^{U}\widehat{m}_{2} \hat{d}_{2},\qquad \hat{c}_{2} h_{3}^{\ell}< c_{2}^{U} \hat{h}_{3}, \\& A_{0}^{+}< u^{+}\triangleq\frac{\hat{r}+\sqrt{\hat{r}^{2}-4\hat {k}\hat{h}_{1}}}{2\hat{k}}, \qquad A_{0}^{-}>u^{-}\triangleq\frac{\hat{r}-\sqrt{\hat{r}^{2}-4\hat {k}\hat{h}_{1}}}{2\hat{k}}, \\& B_{0}^{+}< v^{+} \\& \hphantom{B_{0}^{+}} \triangleq\frac{1}{2\widehat{m}_{1}\hat{d}_{1}} \biggl[ ( \hat{c}_{1}-\hat{d}_{1})\frac{h_{1}^{\ell}}{r^{U}}-\widehat {m}_{1}\hat{h}_{2}+ \sqrt{ \biggl[ ( \hat{c}_{1}-\hat{d}_{1})\frac{h_{1}^{\ell}}{r^{U}}-\widehat {m}_{1}\hat{h}_{2} \biggr] ^{2}-\frac{4r^{U}\widehat{m}_{1} \hat{d}_{1}\hat{h}_{2}}{k^{\ell}}} \biggr] , \\& B_{0}^{-}>v^{-} \\& \hphantom{B_{0}^{-}}\triangleq\frac{1}{2\widehat{m}_{1}\hat{d}_{1}} \biggl[ ( \hat{c}_{1}-\hat{d}_{1})\frac{h_{1}^{\ell}}{r^{U}}-\widehat {m}_{1}\hat{h}_{2}- \sqrt{ \biggl[ ( \hat{c}_{1}-\hat{d}_{1})\frac{h_{1}^{\ell}}{r^{U}}-\widehat {m}_{1}\hat{h}_{2} \biggr] ^{2}-\frac{4r^{U}\widehat{m}_{1} \hat{d}_{1}\hat{h}_{2}}{k^{\ell}}} \biggr] , \\& C_{0}^{+}< w^{+} \\& \hphantom{C_{0}^{+}}\triangleq\frac{1}{2\widehat{m}_{2}\hat{d}_{2}} \biggl[ ( \hat{c}_{2}-\hat{d}_{2})\frac{h_{2}^{\ell}}{c_{1}^{U}}-\widehat {m}_{2}\hat{h}_{3}+ \sqrt{ \biggl[ ( \hat{c}_{2}-\hat{d}_{2})\frac{h_{2}^{\ell}}{c_{1}^{U}}- \widehat{m}_{2}\hat{h}_{3} \biggr] ^{2}- \frac{4r^{U}c_{1}^{U}\widehat{m}_{2} \hat{d}_{2}\hat{h}_{3}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}}} \biggr] , \\& C_{0}^{-}>w^{-} \\& \hphantom{C_{0}^{-}}\triangleq\frac{1}{2\widehat{m}_{2}\hat{d}_{2}} \biggl[ ( \hat{c}_{2}-\hat{d}_{2})\frac{h_{2}^{\ell}}{c_{1}^{U}}-\widehat {m}_{2}\hat{h}_{3}- \sqrt{ \biggl[ ( \hat{c}_{2}-\hat{d}_{2})\frac{h_{2}^{\ell}}{c_{1}^{U}}- \widehat{m}_{2}\hat{h}_{3} \biggr] ^{2}- \frac{4r^{U}c_{1}^{U}\widehat{m}_{2} \hat{d}_{2}\hat{h}_{3}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}}} \biggr] . \end{aligned}$$
(2.25)
To this end, define a mapping \(\phi_{1}:\operatorname{Dom} L\times[0,1]\rightarrow X\) by
$$ \phi_{1}(u,v,w,\mu_{1})=\mu_{1}\left [ \textstyle\begin{array}{@{}c@{}} N_{1}(t_{1})\\ N_{2}(t_{2})\\ N_{3}(t_{3}) \end{array}\displaystyle \right ] +(1-\mu_{1})\left [ \textstyle\begin{array}{@{}c@{}} \widehat{N}_{1}\\ \widehat{N}_{2}\\ \widehat{N}_{3} \end{array}\displaystyle \right ], $$
where \(\mu_{1}\in[0,1]\) is a parameter.
Now we show that \(\phi_{1}(u,v,w,\mu_{1})\neq0\), \((u,v,w)^{T}\in\partial \Omega_{i}\cap \operatorname{Ker} L=\partial\Omega_{i}\cap \mathbb{R}^{3}\), \(i=1,\ldots,8\). If it is not the case, then when \((u,v,w)^{T}\in\partial\Omega_{i}\cap \operatorname{Ker} L=\partial\Omega_{i}\cap \mathbb{R}^{3}\), \(i=1,\ldots,8\), \(\phi_{1}(u,v,w,\mu_{1})=0\). Therefore, the constant vector \((u,v,w)^{T}\in\mathbb{R}^{3}\) satisfies
$$\begin{aligned}& \mu_{1} \biggl[ r(t_{1})-k(t_{1})e^{u}- \frac{b_{1}(t_{1})e^{v}}{m_{1}(t_{1})e^{v}+e^{u}}-\frac {h_{1}(t_{1})}{e^{u}} \biggr] +(1-\mu_{1}) \bigl( \hat{r}-\hat{k}e^{u}-\hat{h}_{1} e^{-u}\bigr)=0, \end{aligned}$$
(2.26)
$$\begin{aligned}& \mu_{1} \biggl[ \frac{c_{1}(t_{2})e^{u}}{m_{1}(t_{2})e^{v}+e^{u}}-d_{1}(t_{2})- \frac {b_{2}(t_{2})e^{w}}{m_{2}(t_{2})e^{w}+e^{v}}- \frac{h_{2}(t_{2})}{e^{v}} \biggr] + (1-\mu_{1}) \\& \quad {}\times \bigl[ (\hat{c}_{1}-\hat{d}_{1}) e^{u+v}-\widehat{m}_{1}\hat{d}_{1} e^{2v}- \hat{h}_{2} e^{u}-\widehat{m}_{1} \hat{h}_{2} e^{v} \bigr] =0, \end{aligned}$$
(2.27)
$$\begin{aligned}& \mu_{1} \biggl[ \frac {c_{2}(t_{3})e^{v}}{m_{2}(t_{3})e^{w}+e^{v}}-d_{2}(t_{3})- \frac {h_{3}(t_{3})}{e^{w}} \biggr] +(1-\mu_{1}) \\& \quad {} \times\bigl[(\hat{c}_{2}-\hat{d}_{2}) e^{v+w}-\hat{h}_{3}\bigl(\widehat{m}_{2} e^{w}+e^{v}\bigr)-\widehat{m}_{2} \hat{d}_{2} e^{2w}\bigr]=0. \end{aligned}$$
(2.28)
From (2.26)-(2.28), we make the following nine claims.
(1) \(u<\ln \{ \frac{r^{U}}{k^{\ell}} \} \). Otherwise, \(u\geq\ln \{ \frac{r^{U}}{k^{\ell}} \} \). Then
$$\begin{aligned}& \mu_{1} \biggl[ r(t_{1})-k(t_{1})e^{u}- \frac{b_{1}(t_{1})e^{v}}{m_{1}(t_{1})e^{v}+e^{u}}-\frac {h_{1}(t_{1})}{e^{u}} \biggr] +(1-\mu_{1}) \bigl( \hat{r}-\hat{k}e^{u}-\hat{h}_{1} e^{-u}\bigr) \\& \quad < \mu_{1}\bigl(r^{U}-k^{\ell}e^{u} \bigr)+(1-\mu_{1}) \bigl(\hat{r}-\hat {k}e^{u}\bigr) \\& \quad < \mu_{1} \biggl(r^{U}-k^{\ell}\frac{r^{U}}{k^{\ell}} \biggr)+(1-\mu _{1}) \biggl( \hat{r}-\hat{k} \frac{r^{U}}{k^{\ell}} \biggr) \\& \quad < 0. \end{aligned}$$
(2) \(u>\ln \{ \frac{h_{1}^{\ell}}{r^{U}} \} \). Otherwise, \(u\leq\ln \{ \frac{h_{1}^{\ell}}{r^{U}} \} \). Then
$$\begin{aligned}& \mu_{1} \biggl[ r(t_{1})-k(t_{1})e^{u}- \frac{b_{1}(t_{1})e^{v}}{m_{1}(t_{1})e^{v}+e^{u}}-\frac {h_{1}(t_{1})}{e^{u}} \biggr] +(1-\mu_{1}) \bigl( \hat{r}-\hat{k}e^{u}-\hat{h}_{1} e^{-u}\bigr) \\& \quad < \mu_{1} \biggl(r^{U}-h_{1}^{\ell}\frac{r^{U}}{h_{1}^{\ell}} \biggr)+(1-\mu_{1}) \biggl( \hat{r}- \hat{h}_{1}\frac{r^{U}}{h_{1}^{\ell}} \biggr) \\& \quad < 0. \end{aligned}$$
(3) \(u>\ln A_{0}^{+}\) or \(u<\ln A_{0}^{-}\). Otherwise, \(\ln A_{0}^{-}\leq u\leq\ln A_{0}^{+}\). Then
$$\begin{aligned}& \mu_{1} \biggl[ r(t_{1})-k(t_{1})e^{u}- \frac{b_{1}(t_{1})e^{v}}{m_{1}(t_{1})e^{v}+e^{u}}-\frac {h_{1}(t_{1})}{e^{u}} \biggr] +(1-\mu_{1}) \bigl( \hat{r}-\hat{k}e^{u}-\hat{h}_{1} e^{-u}\bigr) \\& \quad =\frac{-\mu_{1}}{e^{u}} \biggl[ k(t_{1}) e^{2u}+ \frac{b_{1}(t_{1})e^{v+u}}{m_{1}(t_{1})e^{v}+e^{u}}-r(t_{1}) e^{u}+h_{1}(t_{1}) \biggr] \\& \qquad {}-\frac{1-\mu_{1}}{e^{u}}\bigl(\hat{k}e^{2u}-\hat {r} e^{u} +\hat{h}_{1}\bigr) \\& \quad >\frac{-\mu_{1}}{e^{u}} \biggl[ k^{U} e^{2u}+ \biggl( \frac{b_{1}}{m_{1}} \biggr)^{U} e^{u}-r^{\ell}e^{u}+h_{1}^{U} \biggr] -\frac{1-\mu_{1}}{e^{u}}\bigl( \hat{k}e^{2u}-\hat {r} e^{u} +\hat{h}_{1}\bigr) \\& \quad >-\frac{1-\mu_{1}}{e^{u}}\bigl(\hat{k}e^{2u}-\hat{r} e^{u} +\hat{h}_{1}\bigr) \\& \quad >0. \end{aligned}$$
Clearly, the above three inequalities contradict (2.26). Hence Claims 1-3 hold.
(4) \(v< \{ \frac{r^{U}c_{1}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \} \). Otherwise, \(v\geq \{ \frac{r^{U}c_{1}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \} \). Then
$$\begin{aligned}& \mu_{1} \biggl[ \frac{c_{1}(t_{2})e^{u}}{m_{1}(t_{2})e^{v}+e^{u}}-d_{1}(t_{2})- \frac {b_{2}(t_{2})e^{w}}{m_{2}(t_{2})e^{w}+e^{v}}- \frac{h_{2}(t_{2})}{e^{v}} \biggr] \\& \qquad {}+(1-\mu_{1})\times \bigl[ (\hat{c}_{1}- \hat{d}_{1}) e^{u+v}-\widehat{m}_{1} \hat{d}_{1} e^{2v}-\hat{h}_{2} e^{u}- \widehat{m}_{1}\hat{h}_{2} e^{v} \bigr] \\& \quad < \mu_{1} \biggl[ -d_{1}^{\ell}+ \frac{c_{1}^{U} e^{u}}{m_{1}^{\ell}e^{v}} \biggr] +(1-\mu_{1})e^{2v} \biggl( \frac{\hat{c}_{1} e^{u}}{e^{v}}-\widehat{m}_{1}\hat{d}_{1} \biggr) \\& \quad < (1-\mu_{1})e^{2v} \biggl(\frac{\hat{c}_{1} r^{U} k^{\ell}m_{1}^{\ell}d_{1}^{\ell}}{k^{\ell}r^{U} c_{1}^{U}}- \widehat{m}_{1}\hat{d}_{1} \biggr) \\& \quad < 0. \end{aligned}$$
(5) \(v>\ln \{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \} \). Otherwise, \(v\leq\ln \{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \} \). Then
$$\begin{aligned}& \mu_{1} \biggl[ \frac{c_{1}(t_{2})e^{u}}{m_{1}(t_{2})e^{v}+e^{u}}-d_{1}(t_{2})- \frac {b_{2}(t_{2})e^{w}}{m_{2}(t_{2})e^{w}+e^{v}}- \frac{h_{2}(t_{2})}{e^{v}} \biggr] \\& \qquad {}+ (1-\mu_{1})\times \bigl[ (\hat{c}_{1}- \hat{d}_{1}) e^{u+v}-\widehat{m}_{1} \hat{d}_{1} e^{2v}-\hat{h}_{2} e^{u}- \widehat{m}_{1}\hat{h}_{2} e^{v} \bigr] \\& \quad < \mu_{1} \biggl[ c_{1}^{U}-h_{2}^{\ell}\frac{c_{1}^{U}}{h_{2}^{\ell}} \biggr] +(1-\mu_{1}) \bigl(\widehat{m}_{1} e^{v}+e^{u}\bigr)e^{v} \biggl[ \frac{\hat{c}_{1} e^{u}}{\widehat{m}_{1} e^{v}+e^{u}}- \frac{\hat{h}_{2}}{e^{v}} \biggr] \\& \quad < (1-\mu_{1}) \bigl(\widehat{m}_{1} e^{v}+e^{u}\bigr)e^{v} \biggl[ \hat{c}_{1}-\hat{h}_{2}\frac{c_{1}^{U}}{h_{2}^{\ell}} \biggr] \\& \quad < 0. \end{aligned}$$
(6) \(v>\ln B_{0}^{+}\) or \(v<\ln B_{0}^{-}\). Otherwise, \(\ln B_{0}^{-}\leq v\leq\ln B_{0}^{+}\). Then
$$\begin{aligned}& \mu_{1} \biggl[ \frac{c_{1}(t_{2})e^{u}}{m_{1}(t_{2})e^{v}+e^{u}}-d_{1}(t_{2})- \frac {b_{2}(t_{2})e^{w}}{m_{2}(t_{2})e^{w}+e^{v}}- \frac{h_{2}(t_{2})}{e^{v}} \biggr] \\& \qquad {}+ (1-\mu_{1})\times \bigl[ (\hat{c}_{1}- \hat{d}_{1}) e^{u+v}-\widehat{m}_{1} \hat{d}_{1} e^{2v}-\hat{h}_{2} e^{u}- \widehat{m}_{1}\hat{h}_{2} e^{v} \bigr] \\& \quad >\frac{-\mu_{1}}{e^{v}(m_{1}^{U} e^{v}+e^{u})} \biggl\{ m_{1}^{U} \biggl[ d_{1}^{U}+ \biggl(\frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] e^{2v} \\& \qquad {}- \biggl[ \biggl(c_{1}^{\ell}-d_{1}^{U}- \biggl(\frac{b_{2}}{m_{2}} \biggr)^{U} \biggr)\frac{h_{1}^{\ell}}{r^{U}}-h_{2}^{U}m_{1}^{U} \biggr] e^{v}+\frac{h_{2}^{U}r^{U}}{k^{\ell}} \biggr\} \\& \qquad {}-(1-\mu_{1}) \biggl\{ \widehat{m}_{1} \hat{d}_{1} e^{2v}- \biggl[ (\hat{c}_{1}- \hat{d}_{1})\frac{h_{1}^{\ell}}{r^{U}}-\widehat {m}_{1} \hat{h}_{2} \biggr] e^{v}+\hat{h}_{2} \frac{r^{U}}{k^{\ell}} \biggr\} \\& \quad >0. \end{aligned}$$
It is easy to see the above three inequalities contradict (2.27). Therefore, Claims 4-6 hold.
(7) \(w<\ln \{ \frac{r^{U} c_{1}^{U}c_{2}^{U} }{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \} \). Otherwise, \(w\geq\ln \{ \frac{r^{U} c_{1}^{U}c_{2}^{U} }{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \} \). Then
$$\begin{aligned}& \mu_{1} \biggl[ \frac {c_{2}(t_{3})e^{v}}{m_{2}(t_{3})e^{w}+e^{v}}-d_{2}(t_{3})- \frac {h_{3}(t_{3})}{e^{w}} \biggr] +(1-\mu_{1}) \\& \qquad {}\times\bigl[(\hat{c}_{2}-\hat{d}_{2}) e^{v+w}-\hat{h}_{3}\bigl(\widehat{m}_{2} e^{w}+e^{v}\bigr)-\widehat{m}_{2} \hat{d}_{2} e^{2w}\bigr] \\& \quad < \mu_{1} \biggl[ \frac{r^{U}c_{1}^{U}c_{2}^{U} }{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}e^{w}}-d_{2}^{\ell}\biggr] +(1-\mu_{1})e ^{2w} \biggl[ \frac{(\hat{c}_{2}-\hat{d}_{2})e^{v}}{e^{w}}- \widehat{m}_{2}\hat {d}_{2} \biggr] \\& \quad < (1-\mu_{1})e ^{2w} \biggl[ \frac{r^{U}c_{1}^{U} \hat {c}_{2}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}e^{w}}- \widehat{m}_{2}\hat{d}_{2} \biggr] \\& \quad < 0. \end{aligned}$$
(8) \(w>\ln \{ \frac{h_{3}^{\ell}}{c_{2}^{U}} \} \). Otherwise, \(w\leq\ln \{ \frac{h_{3}^{\ell}}{c_{2}^{U}} \} \). Then
$$\begin{aligned}& \mu_{1} \biggl[ \frac {c_{2}(t_{3})e^{v}}{m_{2}(t_{3})e^{w}+e^{v}}-d_{2}(t_{3})- \frac {h_{3}(t_{3})}{e^{w}} \biggr] +(1-\mu_{1}) \\& \qquad {}\times\bigl[(\hat{c}_{2}-\hat{d}_{2}) e^{v+w}-\hat{h}_{3}\bigl(\widehat{m}_{2} e^{w}+e^{v}\bigr)-\widehat{m}_{2} \hat{d}_{2} e^{2w}\bigr] \\& \quad < \mu_{1} \biggl[ c_{2}^{U}-h_{3}^{\ell}\frac{c_{2}^{U}}{h_{3}^{\ell}} \biggr] +(1-\mu_{1}) \bigl(\widehat{m}_{2} e^{w}+e^{v}\bigr)e^{w} \biggl[ \frac{\hat{c}_{2} e^{v}}{\widehat{m}_{2} e^{w}+e^{v}}- \frac{\hat{h}_{3}}{e^{w}} \biggr] \\& \quad < (1-\mu_{1}) \bigl(\widehat{m}_{2} e^{w}+e^{v}\bigr)e^{w} \biggl[ \hat{c}_{2}-\hat{h}_{3}\frac{c_{2}^{U}}{h_{3}^{\ell}} \biggr] \\& \quad < 0. \end{aligned}$$
(9) \(w>\ln C_{0}^{+}\) or \(w<\ln C_{0}^{-}\). Otherwise, \(\ln C_{0}^{-}\leq w\leq\ln C_{0}^{+}\). Then
$$\begin{aligned}& \mu_{1} \biggl[ \frac {c_{2}(t_{3})e^{v}}{m_{2}(t_{3})e^{w}+e^{v}}-d_{2}(t_{3})- \frac {h_{3}(t_{3})}{e^{w}} \biggr] +(1-\mu_{1}) \\& \qquad {}\times\bigl[(\hat{c}_{2}-\hat{d}_{2}) e^{v+w}-\hat{h}_{3}\bigl(\widehat{m}_{2} e^{w}+e^{v}\bigr)-\widehat{m}_{2} \hat{d}_{2} e^{2w}\bigr] \\& \quad >\frac{-\mu_{1}}{(m_{2}(t_{3})e^{w}+e^{v})e^{w}} \biggl[ d_{2}^{U} m_{2}^{U}e^{2w}+h_{3}^{U} \frac{r^{U}c_{1}^{U} }{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}}- \biggl((c_{2}-d_{2})^{\ell}\frac{h_{2}^{\ell}}{c_{1}^{U}}-m_{2}^{U} h_{3}^{U} \biggr) e^{w} \biggr] \\& \qquad {}-(1-\mu_{1}) \biggl\{ \widehat{m}_{2}\hat{d}_{2} e^{2w}- \biggl[ (\hat{c}_{2}-\hat{d}_{2}) \frac{h_{2}^{\ell}}{c_{1}^{U}}-\widehat{m}_{2}\hat{h}_{3} \biggr] e^{w}+\hat{h}_{3}\frac{r^{U}c_{1}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \biggr\} \\& \quad > -(1-\mu_{1}) \biggl\{ \widehat{m}_{2} \hat{d}_{2} e^{2w}- \biggl[ (\hat{c}_{2}- \hat{d}_{2})\frac{h_{2}^{\ell}}{c_{1}^{U}}-\widehat{m}_{2} \hat{h}_{3} \biggr] e^{w}+\hat{h}_{3} \frac{r^{U}c_{1}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \biggr\} \\& \quad >0. \end{aligned}$$
Obviously, the above three inequalities contradict (2.28). Hence Claims 7-9 hold.
From the above arguments (1)-(9), we have
$$\begin{aligned}& \ln \biggl\{ \frac{h_{1}^{\ell}}{r^{U}} \biggr\} < u< \ln A_{0}^{-} \quad \mbox{or}\quad \ln A_{0}^{+}< u< \ln \biggl\{ \frac{r^{U}}{k^{\ell}} \biggr\} , \\& \ln \biggl\{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \biggr\} < v< \ln B_{0}^{-} \quad \mbox{or}\quad \ln B_{0}^{+}< v< \ln \biggl\{ \frac{r^{U} c_{1}^{U} }{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \biggr\} , \\& \ln \biggl\{ \frac{h_{3}^{U}}{c_{2}^{U}} \biggr\} < w< \ln C_{0}^{-} \quad \mbox{or} \quad \ln C_{0}^{+}< w< \ln \biggl\{ \frac{r^{U}c_{1}^{U}c_{2}^{U} }{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \biggr\} . \end{aligned}$$
These indicate that \((u,v,w)^{T}\) belongs to one of \(\Omega_{i}\cap\mathbb{R}^{3}\), \(i=1,\ldots,8\). This is a contradiction.
On the other hand, we prove that, for \(i=1,\ldots,8\),
$$\begin{aligned}& \deg\bigl\{ (\widehat{N}_{1},\widehat{N}_{2}, \widehat{N}_{3})^{T},\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} \\& \quad =\deg\bigl\{ \bigl[\hat{r}-\hat{k}e^{u}-\hat{h}_{1} e^{-u}, \hat{c}_{1} e^{u+v}-\widehat{m}_{1} \hat{d}_{1} e^{2v}-\hat{h}_{2} e^{u}, \hat{c}_{2} e^{v+w}-\widehat{m}_{2} \hat{d}_{2} e^{2w}-\hat{h}_{3} e^{v} \bigr]^{T}, \\& \qquad \Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} . \end{aligned}$$
(2.29)
To this end, we define a mapping \(\psi_{2}:\operatorname{Dom} L\times[0,1]\rightarrow X\) by
$$ \psi_{2}(u,v,w,\mu_{2})=\left [ \textstyle\begin{array}{@{}c@{}} \hat{r}-\hat{k}e^{u}-\hat{h}_{1}r^{-u} \\ \hat{c}_{1} e^{u+v}-\widehat{m}_{1}\hat{d}_{1} e^{2v}-\hat{h}_{2}e^{u}-\mu_{2}(\widehat{m}_{1}\hat{h}_{2} e^{v}+\hat{d}_{1} e^{u+v}) \\ \hat{c}_{2} e^{v+w}-\widehat{m}_{2}\hat{d}_{2} e^{2w}-\hat{h}_{3}e^{v}-\mu_{2}(\widehat{m}_{2}\hat{h}_{3} e^{w}+\hat{d}_{2} e^{v+w}) \end{array}\displaystyle \right ], $$
where \(\mu_{2}\in[0,1]\) is a parameter. We prove that when \((u,v,w)^{T}\in\partial\Omega_{i}\cap \operatorname{Ker} L=\partial\Omega_{i}\cap\mathbb{R}^{3}\), \(i=1,\ldots,8\), \(\psi _{2}(u,v,w,\mu_{2})\neq(0,0,0)^{T}\). If it is not true, then the constant vector \((u,v,w)^{T}\in \partial\Omega_{i}\cap\mathbb{R}^{3}\), \(i=1,\ldots,8\) satisfies the following equalities:
$$ \textstyle\begin{cases} \hat{r}-\hat{k}e^{u}-\hat{h}_{1} e^{-u}=0, \\ \hat{c}_{1} e^{u+v}-\widehat{m}_{1}\hat{d}_{1} e^{2v}-\hat{h}_{2}e^{u}-\mu_{2}(\widehat{m}_{1}\hat{h}_{2} e^{v}+\hat{d}_{1} e^{u+v})=0, \\ \hat{c}_{2} e^{v+w}-\widehat{m}_{2}\hat{d}_{2} e^{2w}-\hat{h}_{3}e^{v}-\mu_{2}(\widehat{m}_{2}\hat{h}_{3} e^{w}+\hat{d}_{2} e^{v+w})=0. \end{cases} $$
By similar arguments to the above estimation of \((u,v,w)^{T}\), we can obtain
$$\begin{aligned}& \ln \biggl\{ \frac{\hat{h}_{1}}{\hat {r}} \biggr\} < u< \ln u^{-} \quad \mbox{or} \quad \ln u^{+}< u< \ln \biggl\{ \frac{\hat{r}}{\hat{k}} \biggr\} , \\& \ln \biggl\{ \frac{\hat{h}_{2}}{\hat{c}_{1}} \biggr\} < v< \ln v^{-} \quad \mbox{or}\quad \ln v^{+}< v< \ln \biggl\{ \frac{\hat{r} \hat{c}_{1}}{\hat {k}\widehat {m}_{1}\hat{d}_{1}} \biggr\} , \\& \ln \biggl\{ \frac{\hat{h}_{3}}{\hat{c}_{2}} \biggr\} < w< \ln w^{-} \quad \mbox{or} \quad \ln w^{+}< w< \ln \biggl\{ \frac{\hat{r}\hat{c}_{1}\hat {c}_{2}}{\hat {k}\widehat{m}_{1} \widehat{m}_{2}\hat{d}_{1}\hat{d}_{2}} \biggr\} . \end{aligned}$$
Therefore, combined with the conditions in (2.25), it follows that
$$\begin{aligned}& \ln \biggl\{ \frac{h_{1}^{\ell}}{r^{U}} \biggr\} < u< \ln A_{0}^{-} \quad \mbox{or}\quad \ln A_{0}^{+}< u< \ln \biggl\{ \frac{r^{U}}{k^{\ell}} \biggr\} , \\& \ln \biggl\{ \frac{h_{2}^{\ell}}{c_{1}^{U}} \biggr\} < v< \ln B_{0}^{-} \quad \mbox{or}\quad \ln B_{0}^{+}< v< \ln \biggl\{ \frac{r^{U} c_{1}^{U} }{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}} \biggr\} , \\& \ln \biggl\{ \frac{h_{3}^{U}}{c_{2}^{U}} \biggr\} < w< \ln C_{0}^{-} \quad \mbox{or} \quad \ln C_{0}^{+}< w< \ln \biggl\{ \frac{r^{U} c_{1}^{U}c_{2}^{U} }{k^{\ell}m_{1}^{\ell}m_{2}^{\ell}d_{1}^{\ell}d_{2}^{\ell}} \biggr\} , \end{aligned}$$
which implies \((u,v,w)^{T}\) belongs to one of \(\Omega_{i}\), \(i=1,\ldots,8\). This is a contradiction. Hence \(\psi_{2}(u,v,w,\mu_{2})\neq(0,0,0)^{T}\), \((u,v,w)^{T}\in\partial\Omega _{i}\cap \operatorname{Ker} L=\partial\Omega_{i}\cap\mathbb{R}^{3}\), \(i=1,\ldots,8\).
By using homotopy invariance of topological degree and (2.24), (2.29), we have, for \(i=1,\ldots,8\),
$$\begin{aligned}& \deg\bigl\{ JQNx,\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} \\& \quad =\deg\bigl\{ \psi_{1}(u,v,w,1),\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} \\& \quad =\deg\bigl\{ \psi_{1}(u,v,w,0),\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} \\& \quad =\deg\bigl\{ \psi_{2}(u,v,w,1),\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} \\& \quad =\deg\bigl\{ \psi_{2}(u,v,w,0),\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} \\& \quad =\deg\bigl\{ \bigl[\hat{r}-\hat{k}e^{u}-\hat{h}_{1} e^{-u}, \hat{c}_{1} e^{u+v}-\widehat{m}_{1} \hat{d}_{1} e^{2v}-\hat{h}_{2} e^{u},\hat{c}_{2} e^{v+w}-\widehat{m}_{2} \hat{d}_{2} e^{2w}-\hat{h}_{3} e^{v} \bigr]^{T}, \\& \qquad \Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} . \end{aligned}$$
Now, we consider the following algebraic equations:
$$ \textstyle\begin{cases} \hat{r}-\hat{k}e^{u}-\hat{h}_{1}e^{-u}=0, \\ \hat{c}_{1} e^{u+v}-\widehat{m}_{1}\hat{d}_{1} e^{2v}-\hat{h}_{2}e^{u}=0, \\ \hat{c}_{2} e^{v+w}-\widehat{m}_{2}\hat{d}_{2} e^{2w}-\hat{h}_{3}e^{v}=0. \end{cases} $$
It is not difficult to find the equations has eight distinct solutions,
$$\begin{aligned}& \bigl(u_{1}^{*},v_{1}^{*},w_{1}^{*}\bigr)=\bigl(\ln u^{+}, \ln v_{+}^{+}, \ln w_{++}^{+}\bigr),\qquad \bigl(u_{2}^{*},v_{2}^{*},w_{2}^{*} \bigr)=\bigl(\ln u^{+},\ln v_{+}^{+}, \ln w_{++}^{-}\bigr), \\& \bigl(u_{3}^{*},v_{3}^{*},w_{3}^{*}\bigr)=\bigl(\ln u^{+}, \ln v_{-}^{+}, \ln w_{+-}^{+}\bigr),\qquad \bigl(u_{4}^{*},v_{4}^{*},w_{4}^{*} \bigr)=\bigl(\ln u^{+},\ln v_{-}^{+}, \ln w_{+-}^{-}\bigr), \\& \bigl(u_{5}^{*},v_{5}^{*},w_{5}^{*}\bigr)=\bigl(\ln u^{-},\ln v_{+}^{-}, \ln w_{-+}^{+}\bigr),\qquad \bigl(u_{6}^{*},v_{6}^{*},w_{6}^{*} \bigr)=\bigl(\ln u^{-},\ln v_{+}^{-}, \ln w_{-+}^{-}\bigr), \\& \bigl(u_{7}^{*},v_{7}^{*},w_{7}^{*}\bigr)=\bigl(\ln u^{-},\ln v_{-}^{-}, \ln w_{--}^{+}\bigr),\qquad \bigl(u_{8}^{*},v_{8}^{*},w_{8}^{*} \bigr)=\bigl(\ln u^{-},\ln v_{-}^{-}, \ln w_{--}^{-}\bigr), \end{aligned}$$
where
$$\begin{aligned}& v_{+}^{\pm}=\ln \biggl\{ \frac{\hat{c}_{1}u^{+} \pm\sqrt{(\hat{c}_{1}u^{+})^{2}-4\widehat{m}_{1}\hat{d}_{1}\hat {h}_{2}u^{+}}}{2\widehat{m}_{1}\hat{d}_{1}} \biggr\} , \\& v_{-}^{\pm}=\ln \biggl\{ \frac{\hat{c}_{1}u^{-} \pm\sqrt{(\hat{c}_{1}u^{-})^{2}-4\widehat{m}_{1}\hat{d}_{1}\hat {h}_{2}u^{-}}}{2\widehat{m}_{1}\hat{d}_{1}} \biggr\} , \\& w_{++}^{\pm}=\ln \biggl\{ \frac{\hat{c}_{2} v_{+}^{+}\pm \sqrt{(\hat{c}_{2}v_{+}^{+})^{2}-4\widehat{m}_{2}\hat{d}_{2}\hat {h}_{3}v_{+}^{+}}}{ 2\widehat{m}_{2}\hat{d}_{2}} \biggr\} , \\& w_{-+}^{\pm}=\ln \biggl\{ \frac{\hat{c}_{2} v_{+}^{-}\pm \sqrt{(\hat{c}_{2}v_{+}^{-})^{2}-4\widehat{m}_{2}\hat{d}_{2}\hat {h}_{3}v_{+}^{-}}}{ 2\widehat{m}_{2}\hat{d}_{2}} \biggr\} , \\& w_{+-}^{\pm}=\ln \biggl\{ \frac{\hat{c}_{2} v_{-}^{+}\pm \sqrt{(\hat{c}_{2}v_{-}^{+})^{2}-4\widehat{m}_{2}\hat{d}_{2}\hat {h}_{3}v_{-}^{+}}}{ 2\widehat{m}_{2}\hat{d}_{2}} \biggr\} , \\& w_{--}^{\pm}=\ln \biggl\{ \frac{\hat{c}_{2} v_{-}^{-}\pm \sqrt{(\hat{c}_{2}v_{-}^{-})^{2}-4\widehat{m}_{2}\hat{d}_{2}\hat {h}_{3}v_{-}^{-}}}{ 2\widehat{m}_{2}\hat{d}_{2}} \biggr\} . \end{aligned}$$
It is easy to verify that \((u_{i}^{*},v_{i}^{*},w_{i}^{*})\) belongs to one of \(\Omega_{j}\), \(i,j=1,\ldots,8\).
It follows from the definition of the topological degree that
$$\begin{aligned}& \deg\bigl\{ JQNx,\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} \\& \quad =\operatorname{sign}\left \vert \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} -\hat{k}e^{u_{i}^{*}}+\hat{h}_{1} e^{-u_{i}^{*}} &0&0\\ \hat{c}_{1} e^{u_{i}^{*}+v_{i}^{*}}-\hat{h}_{2} e^{u_{i}^{*}} &\hat{c}_{1} e^{u_{i}^{*}+v_{i}^{*}}-2\widehat{m}_{1}\hat{d}_{1} e^{2v} &0\\ 0 &\hat{c}_{2} e^{v_{i}^{*}+w_{i}^{*}}-\hat{h}_{3} e^{v_{i}^{*}} & \hat{c}_{2}e^{v_{i}^{*}+w_{i}^{*}}-2\widehat{m}_{2}\hat{d}_{2} e^{2w_{i}^{*}} \end{array}\displaystyle \right \vert \\& \quad =\operatorname{sign} \bigl[ \bigl(-\hat{k}e^{u_{i}^{*}}+ \hat{h}_{1} e^{-u_{i}^{*}} \bigr) \bigl(\hat{c}_{1} e^{u_{i}^{*}+v_{i}^{*}}-2\widehat{m}_{1}\hat{d}_{1} e^{2v_{i}^{*}} \bigr) \bigl(\hat{c}_{2}e^{v_{i}^{*}+w_{i}^{*}}-2\widehat {m}_{2} \hat{d}_{2} e^{2w_{i}^{*}} \bigr) \bigr] \\& \quad =-\operatorname{sign} \bigl[ \bigl(2\hat{k}e^{u_{i}^{*}}-\hat{r} \bigr) \bigl( \hat{c}_{1}e^{u_{i}^{*}}-2\widehat{m}_{1} \hat{d}_{1} e^{v_{i}^{*}} \bigr) \bigl(\hat{c}_{2}e^{v_{i}^{*}}-2 \widehat{m}_{2}\hat{d}_{2} e^{w_{i}^{*}} \bigr) \bigr] . \end{aligned}$$
Then, by direct calculation, we obtain
$$ \deg\bigl\{ JQNx,\Omega_{i}\cap \operatorname{Ker} L,(0,0,0)^{T}\bigr\} \neq0,\quad i=1,\ldots,8. $$
By now, we have proved that each \(\Omega_{i}\) (\(i=1,\ldots,8\)) satisfies all the requirements of Theorem A. Hence, system (2.1) has at least one ω-periodic solution in each of \(\Omega_{1},\ldots,\Omega_{8}\). The proof is completed. □

Theorem 2.2

If \(h_{1}(t)\neq0\), \(h_{2}(t)\neq0\), \(h_{3}(t)=0\), and (H1), (H2) are satisfied. Moreover,
$$(\mathrm{H}4)\quad c_{2}^{\ell}>d_{2}^{U}. $$
Then system (1.3) has at least four positive periodic solutions.

Theorem 2.3

If \(h_{1}(t)\neq0\), \(h_{2}(t)=0\), \(h_{3}(t)\neq0\), and (H1) is satisfied. Moreover,
$$\begin{aligned} (\mathrm{H}5)&\quad c_{1}^{\ell}>d_{1}^{U}+ \biggl(\frac{b_{2}}{m_{2}} \biggr)^{U} , \\ (\mathrm{H}6)&\quad \bigl(c_{2}^{\ell}-d_{2}^{U} \bigr) \frac{[c_{1}^{\ell}-d_{1}^{U}-(b_{2}/{m_{2}})^{U}]h_{1}^{\ell}}{r^{U}m_{1}^{U}[d_{1}^{U}+(b_{2}/{m_{2}})^{U}]}-m_{2}^{U} h_{3}^{U}>2 \sqrt{\frac{r^{U} c_{1}^{U} m_{2}^{U} d_{2}^{U} h_{3}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}}} . \end{aligned}$$
Then system (1.3) has at least four positive periodic solutions.

Theorem 2.4

If \(h_{1}(t)=0\), \(h_{2}(t)\neq0\), \(h_{3}(t)\neq0\), and (H3) is satisfied. Moreover,
$$\begin{aligned} (\mathrm{H}7)&\quad r^{\ell}> \biggl(\frac{b_{1}}{m_{1}} \biggr)^{U} , \\ (\mathrm{H}8) &\quad \biggl[ c_{1}^{\ell}-d_{1}^{U}- \biggl(\frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] \frac{r^{\ell}-(b_{1}/{m_{1}})^{U}}{k^{U}}-m_{1}^{U}h_{2}^{U}>2 \sqrt{m_{1}^{U} \biggl[ d_{1}^{U}+ \biggl(\frac{b_{2}}{m_{2}} \biggr)^{U} \biggr] \frac{r^{U} h_{2}^{U} }{k^{\ell}}} . \end{aligned}$$
Then system (1.3) has at least four positive periodic solutions.

Theorem 2.5

If \(h_{1}(t)\neq0\), \(h_{2}(t)=0\), \(h_{3}(t)=0\), and (H1), (H4), (H5) are satisfied, then system (1.3) has at least two positive periodic solutions.

Theorem 2.6

If \(h_{1}(t)=0\), \(h_{2}(t)\neq0\), \(h_{3}(t)=0\), and (H4), (H7), (H8) are satisfied, then system (1.3) has at least two positive periodic solutions.

Theorem 2.7

If \(h_{1}(t)=0\), \(h_{2}(t)=0\), \(h_{3}(t)\neq0\), and (H5), (H7) are satisfied. Moreover,
$$(\mathrm{H}9)\quad \bigl(c_{2}^{\ell}-d_{2}^{U} \bigr) \frac{[c_{1}^{\ell}-d_{1}^{U}-(b_{2}/{m_{2}})^{U}][r^{\ell}-(b_{1}/{m_{1}})^{U}]}{k^{U} m_{1}^{U}[d_{1}^{U}+(b_{2}/{m_{2}})^{U}]}-m_{2}^{U} h_{3}^{U}>2 \sqrt{\frac{r^{U} c_{1}^{U} m_{2}^{U} d_{2}^{U} h_{3}^{U}}{k^{\ell}m_{1}^{\ell}d_{1}^{\ell}}}. $$
Then system (1.3) has at least two positive periodic solutions.

Theorem 2.8

If \(h_{1}(t)=0\), \(h_{2}(t)=0\), \(h_{3}(t)=0\), and (H4), (H5), (H7) are satisfied, then system (1.3) has at least one positive periodic solution.

3 Conclusions

In this paper, with the help of a continuation theorem based on Gaines and Mawhin’s coincidence degree theory, we study the existence and multiplicity of periodic solutions of a ratio-dependent food chain model with exploited term(s). Under some appropriate conditions, some sufficient criteria are established for the existence and multiplicity of periodic solutions. It worth mentioning that the results reported here are rather interesting. To make this point clear, we take i= number of exploited terms, then by our main results, there are at least \(2^{i}\) periodic solutions. In fact, by our observation, the same result is valid for the models with one prey and one predator in the literature; for example, see [6, 7, 11, 13]. So, a natural question that one may ask is whether the assertion is fit for higher-dimensional biological and ecological systems (≥4).

Declarations

Acknowledgements

The authors would like to thank the referees for their valuable suggestions and comments. This work was supported by NSFC (Nos. 11101075, 11271065).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, P.R. China

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Copyright

© Zeng 2016

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