## 1 Introduction

The past decades have witnessed an enormous interest in predator-prey systems, with most of them focusing on homogeneous populations [1–5]. In fact, no natural population is truly homogeneous, many organisms undergo radical changes in many aspects such as the rates of survival, maturation, reproduction and predation while while they are processing their life history. Therefore, stage-structured predator-prey systems have received much attention recently [6–21].

However, these works [7–21] are largely using constant coefficient systems and have stage structures on only one of the interactive species. The research objectives mainly include the stability and permanence of the system, often with one particular type of functional response. In this article, we propose the following variable coefficient predator-prey system with time delays and stage structures for both interactive species, as well as different functional responses, Specifically, we will consider the condition for the permanence of the two species and the non-permanence condition for predators to become extinct:

$$\begin{array}{c}{\dot{x}}_{1}(t)={r}_{1}(t){x}_{2}(t)-{d}_{11}{x}_{1}(t)-{r}_{1}(t){e}^{-{d}_{11}{\tau}_{1}}{x}_{2}(t-{\tau}_{1})\hfill \\ {\dot{x}}_{2}(t)={r}_{1}(t){e}^{-{d}_{11}{\tau}_{1}}{x}_{2}(t-{\tau}_{1})-{d}_{12}{x}_{2}(t)-{b}_{1}(t){x}_{2}^{2}-{c}_{1}(t)\varphi ({x}_{2}(t)){x}_{2}(t){y}_{2}(t)\hfill \\ {\dot{y}}_{1}(t)={r}_{2}(t){y}_{2}(t)-{d}_{22}{y}_{1}(t)-{r}_{2}(t){e}^{-{d}_{22}{\tau}_{2}}{y}_{2}(t-{\tau}_{2})\hfill \\ {\dot{y}}_{2}(t)={r}_{2}(t){e}^{-{d}_{22}{\tau}_{2}}{y}_{2}(t-{\tau}_{2})-{d}_{21}{y}_{2}(t)-{b}_{2}(t){y}_{2}^{2}+{c}_{2}(t)\varphi ({x}_{2}(t)){x}_{2}(t){y}_{2}(t)\hfill \end{array}$$ | (1) |

_{1}(t) and x

_{2}(t) denote the immature and mature densities of the prey at time t, respectively. y

_{1}(t) and y

_{2}(t) denote the immature and mature densities of the predator at time t, respectively. r

_{i}(t), b

_{i}(t), c

_{i}(t) (i = 1, 2) are positive and continuous functions for all t ≥ 0.

The above system assumes that the mature predators only feed on the mature prey population, the immature are produced by the mature populations. The birth rate of prey and predators (r_{i}(t) > 0; i = 1, 2) is proportional to the existing mature population sizes.

The other parameters have the following biological meanings: c_{1}(t) denotes the capturing rate of mature predators at time t, and c_{2}(t)/c_{1}(t) is the rate of conversion of nutrients from the mature prey into the reproduction of the mature predator; b_{i}(t) > 0 (i = 1, 2) represents the intraspecific competition rate of mature prey and predators at time t, respectively; d_{11} and d_{12} are the death rate of immature and mature population of prey respectively, d_{22} and d_{21} are the death rate of immature and mature population of predator respectively; τ_{i} > 0(i = 1, 2) is the length of time from the birth to maturity of the species i. The term r_{1}(t)e^{−d11τ1}x_{2}(t − τ_{1}) represents the survived prey population born at time t − τ_{1}, that is, the transition from the immature to mature prey. The term r_{2}(t)e^{−d22τ2}y_{2}(t − τ_{2}) denotes the survived predator population born at time t − τ_{2}, that is, the transition from the immature to mature predator.

The term φ(x_{2})x_{2}, the number of prey captured per predator per unit time, is called the predator functional response and satisfies the following assumptions

$$0<\varphi ({x}_{2})<L<+\infty \text{,}\phantom{\rule{1em}{0ex}}\frac{\mathrm{d}}{\mathrm{d}{x}_{2}}(\varphi ({x}_{2}){x}_{2})\ge 0({x}_{2}>0)\text{.}$$ | (2) |

The initial conditions of system (1) are given by

$${x}_{i}(\theta )={\varphi}_{i}(\theta )>0\text{,}\phantom{\rule{1em}{0ex}}{y}_{i}(\theta )={\psi}_{i}(\theta )>0\text{,}\phantom{\rule{1em}{0ex}}{\varphi}_{i}(0)>0\text{,}\phantom{\rule{1em}{0ex}}{\psi}_{i}(0)>0\phantom{\rule{1em}{0ex}}(i=1\text{,}2)\text{,}\phantom{\rule{1em}{0ex}}\theta \in [-\tau \text{,}0]\text{,}\phantom{\rule{1em}{0ex}}\tau =\mathrm{max}\{{\tau}_{1}\text{,}{\tau}_{2}\}\text{.}$$ | (3) |

For the continuity of initial conditions, we require further that

$${x}_{1}(0)={\int}_{-{\tau}_{1}}^{0}{r}_{1}(s){\varphi}_{2}(s)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}s\text{,}\phantom{\rule{1em}{0ex}}{y}_{1}(0)={\int}_{-{\tau}_{2}}^{0}{r}_{2}(s){\psi}_{2}(s)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}s\text{.}$$ | (4) |

Again, we define that

$$\begin{array}{c}{r}_{k}^{s}=\underset{t\ge 0}{\mathrm{sup}}{r}_{k}(t)>0\text{,}{c}_{k}^{s}=\underset{t\ge 0}{\mathrm{sup}}{c}_{k}(t)>0\text{,}{b}_{k}^{s}=\underset{t\ge 0}{\mathrm{sup}}{b}_{k}(t)>0\text{,}\hfill \\ {r}_{k}^{i}=\underset{t\ge 0}{\mathrm{inf}}{r}_{k}(t)>0\text{,}{c}_{k}^{i}=\underset{t\ge 0}{\mathrm{inf}}{c}_{k}(t)>0\text{,}{b}_{k}^{i}=\underset{t\ge 0}{\mathrm{inf}}{b}_{k}(t)>0\phantom{\rule{1em}{0ex}}(k=1\text{,}2)\text{.}\hfill \end{array}$$ | (5) |

## 2 Positivity and boundedness

#### Theorem 2.1

Solutions of system(1)with initial conditions(3) and (4)are positive and bounded for all t ≥ 0.

#### Proof

First we show x_{2}(t) > 0 for all t ≥ 0. Otherwise, noticing that x_{2}(t) = φ_{2}(t) > 0 for all −τ_{1} < t < 0.

Then there exists a t^{∗} > 0, such that x_{2}(t^{∗}) = 0.

Now denoting t_{0} = inf {t > 0 | x_{2}(t) = 0}.

Then t_{0} > 0 and from system (1), we have

$$\begin{array}{c}0\le {t}_{0}\le {\tau}_{1}\Rightarrow {\dot{x}}_{2}({t}_{0})={r}_{1}({t}_{0}){e}^{-{d}_{11}{\tau}_{1}}{\varphi}_{2}({t}_{0}-{\tau}_{1})>0\text{.}\hfill \\ {t}_{0}>{\tau}_{1}\Rightarrow {\dot{x}}_{2}({t}_{0})={r}_{1}({t}_{0}){e}^{-{d}_{11}{\tau}_{1}}{x}_{2}({t}_{0}-{\tau}_{1})>0\text{.}\hfill \end{array}$$ |

_{0}, we have ${\dot{x}}_{2}({t}_{0})\le 0$. A contradiction.

Thus x_{2}(t) > 0 for all t > 0.

Again, considering the following equation

$$\begin{array}{c}\dot{u}(t)=-{d}_{11}u(t)-{r}_{1}(t){e}^{-{d}_{11}{\tau}_{1}}u(t-{\tau}_{1})\hfill \\ u(0)={x}_{2}(0)\hfill \end{array}$$ | (6) |

We can easily obtain that

$$u(t)={e}^{-{d}_{11}t}\left[{\varphi}_{2}(0)-{\int}_{0}^{t}{r}_{1}(s){\varphi}_{2}(s)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}s\right]\text{.}$$ |

$${x}_{1}(t)>u(t)(0\le t<{\tau}_{1})\text{.}$$ |

_{1}) = 0, then x

_{1}(t) > u(t) > 0 for 0 ≤ t < τ

_{1}.

By induction, we can show x_{1}(t) > 0 for all t ≥ 0.

Similarly, we can prove that y_{1}(t) > 0 and y_{2}(t) > 0 for all t ≥ 0.

Next, we will prove the boundedness of the solutions of system (1).

Defining the function

$$V(t)={c}_{2}^{s}{x}_{1}(t)+{c}_{2}^{s}{x}_{2}(t)+{c}_{1}^{i}{y}_{1}(t)+{c}_{1}^{i}{y}_{2}(t)\text{.}$$ |

$$\begin{array}{c}\dot{V}(t)={c}_{2}^{s}{\dot{x}}_{1}(t)+{c}_{2}^{s}{\dot{x}}_{2}(t)+{c}_{1}^{i}{\dot{y}}_{1}(t)+{c}_{1}^{i}{\dot{y}}_{2}(t)\hfill \\ \le ({c}_{2}^{s}{r}_{1}^{s}-{c}_{2}^{s}{d}_{12}){x}_{2}(t)-{c}_{2}^{s}{b}_{1}^{i}{x}_{2}^{2}(t)-{c}_{2}^{s}{d}_{11}{x}_{1}(t)+({c}_{1}^{i}{r}_{2}^{s}-{c}_{1}^{i}{d}_{21}){y}_{2}(t)-{c}_{1}^{i}{b}_{2}^{i}{y}_{2}^{2}(t)-{c}_{1}^{i}{d}_{22}{y}_{1}(t)\text{.}\hfill \end{array}$$ |

_{11}, d

_{22}}, then

$$\dot{V}(t)+dV(t)\le {c}_{2}^{s}({r}_{1}^{s}-{d}_{12}+d){x}_{2}(t)-{c}_{2}^{s}{b}_{1}^{i}{x}_{2}^{2}(t)+{c}_{1}^{i}({r}_{2}^{s}-{c}_{1}^{i}{d}_{21}+d){y}_{2}(t)-{c}_{1}^{i}{b}_{2}^{i}{y}_{2}^{2}(t)\text{.}$$ |

$$V(t)\le M{d}^{-1}+(V(0)-M{d}^{-1}){e}^{-dt}\text{.}$$ |

## 3 Permanence and non-permanence

#### Definition 3.1

If there exist positive constants m_{x}, M_{x}, m_{y} and M_{y}, such that each solution (x_{1}(t), x_{2}(t), y_{1}(t), y_{2}(t)) of system (1) satisfies

$$\begin{array}{c}0<{m}_{x}\le {\mathrm{liminf}}_{t\to +\infty}{x}_{i}(t)\le {\mathrm{limsup}}_{t\to +\infty}{x}_{i}(t)\le {M}_{x}\phantom{\rule{1em}{0ex}}(i=1\text{,}2)\text{,}\hfill \\ 0<{m}_{y}\le {\mathrm{liminf}}_{t\to +\infty}{y}_{i}(t)\le {\mathrm{limsup}}_{t\to +\infty}{y}_{i}(t)\le {M}_{y}\phantom{\rule{1em}{0ex}}(i=1\text{,}2)\text{.}\hfill \end{array}$$ |

#### Lemma 3.2 [22]

Consider the following equation:

$$\dot{u}(t)=au(t-\tau )-bu(t)-c{u}^{2}(t)$$ |

- (1). If a > b, then ${\mathrm{lim}}_{t\to +\infty}u(t)=\frac{a-b}{c}$,
- (2). If a < b, then lim
_{t→+∞}u(t) = 0.

#### Theorem 3.3

If the following assumptions hold

$$\begin{array}{c}({H}_{1})\text{.}\phantom{\rule{1em}{0ex}}{r}_{1}^{i}{x}_{2}^{i}>{r}_{1}^{s}{x}_{2}^{s}{e}^{-{d}_{11}{\tau}_{1}}>0\hfill \\ ({H}_{2})\text{.}\phantom{\rule{1em}{0ex}}{r}_{2}^{i}{y}_{2}^{i}>{r}_{2}^{s}{y}_{2}^{s}{e}^{-{d}_{22}{\tau}_{2}}>0\hfill \end{array}$$ | (7) |

$$\begin{array}{c}{x}_{2}^{i}=\frac{{r}_{1}^{i}{e}^{-{d}_{11}{\tau}_{1}}-{d}_{12}-{c}_{1}^{s}L{y}_{2}^{s}}{{b}_{1}^{s}}\text{,}\phantom{\rule{1em}{0ex}}{x}_{2}^{s}=\frac{{r}_{1}^{s}{e}^{-{d}_{11}{\tau}_{1}}-{d}_{12}}{{b}_{1}^{i}}\text{,}\hfill \\ {y}_{2}^{i}=\frac{{r}_{2}^{i}{e}^{-{d}_{22}}+{c}_{2}^{i}\varphi ({x}_{2}^{i}){x}_{2}^{i}-{d}_{21}}{{b}_{2}^{s}}\text{,}\phantom{\rule{1em}{0ex}}{y}_{2}^{s}=\frac{{r}_{2}^{s}{e}^{-{d}_{22}{\tau}_{2}}+{c}_{2}^{s}\varphi ({x}_{2}^{s}){x}_{2}^{s}-{d}_{21}}{{b}_{2}^{i}}\text{.}\hfill \end{array}$$ |

#### Proof

Let (x_{1}(t), x_{2}(t), y_{1}(t), y_{2}(t)) be any positive solution of system (1) for t ≥ 0. Firstly, from the second equation of system (1), we have

$$\begin{array}{c}{\dot{x}}_{2}(t)\le {r}_{1}(t){e}^{-{d}_{11}{\tau}_{1}}{x}_{2}(t-{\tau}_{1})-{d}_{12}{x}_{2}(t)-{b}_{1}(t){x}_{2}^{2}\hfill \\ \le {r}_{1}^{s}{e}^{-{d}_{11}{\tau}_{1}}{x}_{2}(t-{\tau}_{1})-{d}_{12}{x}_{2}(t)-{b}_{1}^{i}{x}_{2}^{2}(t)\text{.}\hfill \end{array}$$ |

$${\mathrm{limsup}}_{t\to +\infty}{x}_{2}(t)\le \frac{{r}_{1}^{s}{e}^{-{d}_{11}{\tau}_{1}}-{d}_{12}}{{b}_{1}^{i}}\u0117{x}_{2}^{s}>0\text{.}$$ |

_{1}> 0, for any t > T

_{1}> 0, such that ${x}_{2}(t)<{x}_{2}^{s}+\varepsilon $.

Again, from the fourth equation of system (1), there exists a T_{2} > T_{1} > 0, for any t > T_{2}, we have

$$\begin{array}{c}{\dot{y}}_{2}(t)\le {r}_{2}(t){e}^{-{d}_{22}{\tau}_{2}}{y}_{2}(t-{\tau}_{2})-{d}_{21}{y}_{2}(t)-{b}_{2}(t){y}_{2}^{2}+{c}_{2}(t)\varphi ({x}_{2}^{s}+\varepsilon )({x}_{2}^{s}+\varepsilon ){y}_{2}(t)\hfill \\ \le {r}_{2}^{s}{e}^{-{d}_{22}{\tau}_{2}}{y}_{2}(t-{\tau}_{2})-{d}_{21}{y}_{2}(t)-{b}_{2}^{i}{y}_{2}^{2}+{c}_{2}^{s}\varphi ({x}_{2}^{s}+\varepsilon )({x}_{2}^{s}+\varepsilon ){y}_{2}(t)\text{.}\hfill \end{array}$$ |

$${\mathrm{limsup}}_{t\to +\infty}{y}_{2}(t)\le \frac{{r}_{2}^{s}{e}^{-{d}_{22}{\tau}_{2}}+{c}_{2}^{s}\varphi ({x}_{2}^{s}+\varepsilon )({x}_{2}^{s}+\varepsilon )-{d}_{21}}{{b}_{2}^{i}}\text{.}$$ |

$${\mathrm{limsup}}_{t\to +\infty}{y}_{2}(t)\le \frac{{r}_{2}^{s}{e}^{-{d}_{22}{\tau}_{2}}-{d}_{21}+{c}_{2}^{s}\varphi ({x}_{2}^{s}){x}_{2}^{s}}{{b}_{2}^{i}}\u0117{y}_{2}^{s}>0\text{.}$$ |

_{3}> T

_{2}> 0, for any t > T

_{3}> 0, such that ${y}_{2}(t)<{y}_{2}^{s}+\varepsilon $.

Now, substituting it into the second equation of system (1), we have

$$\begin{array}{c}{\dot{x}}_{2}(t)\ge {r}_{1}(t){e}^{-{d}_{11}{\tau}_{1}}{x}_{2}(t-{\tau}_{1})-{d}_{12}{x}_{2}(t)-{b}_{1}(t){x}_{2}^{2}(t)-{c}_{1}(t)({y}_{2}^{s}+\varepsilon )\varphi ({x}_{2}(t)){x}_{2}(t)\hfill \\ \ge {r}_{1}^{i}{e}^{-{d}_{11}{\tau}_{1}}{x}_{2}(t-{\tau}_{1})-{d}_{12}{x}_{2}(t)-{b}_{1}^{s}{x}_{2}^{2}(t)-{c}_{1}^{s}L({y}_{2}^{s}+\varepsilon ){x}_{2}(t)\text{.}\hfill \end{array}$$ |

$${\mathrm{liminf}}_{t\to +\infty}{x}_{2}(t)\ge \frac{{r}_{1}^{i}{e}^{-{d}_{11}{\tau}_{1}}-{d}_{12}-{c}_{1}^{s}L{y}_{2}^{s}}{{b}_{1}^{s}}\u0117{x}_{2}^{i}>0\text{.}$$ |

_{4}> T

_{3}> 0, for any t > T

_{4}> 0, such that ${x}_{2}(t)\ge {x}_{2}^{i}-\varepsilon $.

From the fourth equation of system (1), there exists a T_{5} > T_{4} > 0, for any t > T_{5} > 0, we have

$$\begin{array}{c}{\dot{y}}_{2}(t)\ge {r}_{2}(t){e}^{-{d}_{22}{\tau}_{2}}{y}_{2}(t-{\tau}_{2})-{d}_{21}{y}_{2}(t)-{b}_{2}(t){y}_{2}^{2}(t)+{c}_{2}(t)\varphi ({x}_{2}^{i}-\varepsilon )({x}_{2}^{i}-\varepsilon ){y}_{2}(t)\hfill \\ \ge {r}_{2}^{i}{e}^{-{d}_{22}{\tau}_{2}}{y}_{2}(t-{\tau}_{2})-{d}_{21}{y}_{2}(t)-{b}_{2}^{s}{y}_{2}^{2}(t)+{c}_{2}^{i}\varphi ({x}_{2}^{i}-\varepsilon )({x}_{2}^{i}-\varepsilon ){y}_{2}(t)\text{.}\hfill \end{array}$$ |

$${\mathrm{liminf}}_{t\to +\infty}{y}_{2}(t)\ge \frac{{r}_{2}^{i}{e}^{-{d}_{22}}-{d}_{21}+{c}_{2}^{i}\varphi ({x}_{2}^{i}){x}_{2}^{i}}{{b}_{2}^{s}}\u0117{y}_{2}^{i}>0\text{.}$$ |

_{6}> T

_{5}> 0, for any t > T

_{6}> 0, such that ${y}_{2}(t)\ge {y}_{2}^{i}-\varepsilon $.

Again, from the first equation of system (1), we have

$$\begin{array}{c}{\dot{x}}_{1}(t)\le {r}_{1}(t)({x}_{2}^{s}+\varepsilon )-{d}_{11}{x}_{1}(t)-{r}_{1}(t){e}^{-{d}_{11}{\tau}_{1}}({x}_{2}^{i}-\varepsilon )\hfill \\ \le {r}_{1}^{s}({x}_{2}^{s}+\varepsilon )-{d}_{11}{x}_{1}(t)-{r}_{1}^{i}{e}^{-{d}_{11}{\tau}_{1}}({x}_{2}^{i}-\varepsilon )\text{.}\hfill \end{array}$$ |

$${\mathrm{limsup}}_{t\to +\infty}{x}_{1}(t)\le \frac{{r}_{1}^{s}{x}_{2}^{s}-{r}_{1}^{i}{e}^{-{d}_{11}{\tau}_{1}}{x}_{2}^{i}}{{d}_{11}}\u0117{x}_{1}^{s}>0\text{.}$$ |

_{6}, and by Lemma 3.2, we have

$${\mathrm{limsup}}_{t\to +\infty}{y}_{1}(t)\le \frac{{r}_{2}^{s}{y}_{2}^{s}-{r}_{2}^{i}{e}^{-{d}_{22}{\tau}_{2}}{y}_{2}^{i}}{{d}_{22}}\u0117{y}_{1}^{s}>0\text{.}$$ |

$$\begin{array}{c}{\mathrm{liminf}}_{t\to +\infty}{x}_{1}(t)\ge \frac{{r}_{1}^{i}{x}_{2}^{i}-{r}_{1}^{s}{e}^{-{d}_{11}{\tau}_{1}}{x}_{2}^{s}}{{d}_{11}}\u0117{x}_{1}^{i}>0\text{.}\hfill \\ {\mathrm{liminf}}_{t\to +\infty}{y}_{1}(t)\ge \frac{{r}_{2}^{i}{y}_{2}^{i}-{r}_{2}^{s}{e}^{-{d}_{22}{\tau}_{2}}{y}_{2}^{s}}{{d}_{22}}\u0117{y}_{1}^{i}>0\text{.}\hfill \end{array}$$ |

#### Theorem 3.4

If (H_{1}) and the following assumption hold

$$({H}_{3})\text{.}\phantom{\rule{1em}{0ex}}{r}_{2}^{s}{e}^{-{d}_{22}{\tau}_{2}}+{c}_{2}^{s}\varphi ({x}_{2}^{s}){x}_{2}^{s}<{d}_{21}$$ | (8) |

$${x}_{2}^{s}=\frac{{r}_{1}^{s}{e}^{-{d}_{11}{\tau}_{1}}-{d}_{12}}{{b}_{1}^{i}}\text{.}$$ |

#### Proof

Let (x_{1}(t), x_{2}(t), y_{1}(t), y_{2}(t)) be any positive solution of system (1) for t ≥ 0. In order to prove the non-permanence of predators and permanence of prey population, we only show that

$$\underset{t\to +\infty}{\mathrm{lim}}{y}_{i}(t)=0\phantom{\rule{1em}{0ex}}(i=1\text{,}2)\text{.}$$ |

_{3}) holds, then we obtain that ${y}_{2}^{s}<0$ and ${y}_{2}^{i}<0$.

From the Proof of Theorem 3.3, there exists a T_{7} > T_{6} > 0, for any t > T_{7}, we have

$$\begin{array}{c}{\mathrm{limsup}}_{t\to +\infty}{y}_{2}(t)\le 0\text{,}\hfill \\ {\mathrm{liminf}}_{t\to +\infty}{y}_{2}(t)\ge 0\text{.}\hfill \end{array}$$ | (9) |

$$\underset{t\to +\infty}{\mathrm{lim}}{y}_{2}(t)=0\text{.}$$ |

$$-{d}_{22}{y}_{1}(t)\le {y}_{1}^{\prime}(t)\le -{d}_{22}{y}_{1}(t)\text{.}$$ |

$$0<{y}_{1}(o){e}^{-{d}_{22}t}\le {y}_{1}(t)\le {y}_{1}(o){e}^{-{d}_{22}t}\text{.}$$ |

$$\underset{t\to +\infty}{\mathrm{lim}}{y}_{1}(t)=0\text{.}$$ |

## 4 Discussion

In this article, we have considered a stage-structured (for both interactive populations) predator-prey system with a class of functional response incorporating discrete time delay, and obtain the sufficient conditions for the permanence of system (1) and the non-permanence of predators. Our work has provided some valuable suggestions for regulating populations and saving endangered species. We see that the predators only forage on mature prey as food resource. Consequently, when the prey population size is lower than a certain level predators will not sustain. This is especially true for mature predators which are responsible for recruitment. In other words, as long as the population sizes of prey and predators maintain at a certain positive level, the predator-prey interaction will continue.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China (No. 30970478,31100306).