# A non-autonomous impulsive food-chain model with delays

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## Abstract

A non-autonomous almost periodic prey–predator system with impulsive effects and multiple delays is proposed in this paper, Holling’s type-IV systems and ratio-dependent functional responses are also involved in the model. By applying absolute inequalities, integral inequalities, differential inequalities and the mean-value theorem and other mathematical analysis techniques, we obtain some sufficient conditions which guarantee the permanence of the system. Moreover, we obtain the existence and the uniqueness of the almost periodic solution which is uniformly asymptotically stable by constructing a series of Lyapunov functionals. Finally, we present several numerical examples to verify the theoretical results and present some discussions of pest management in the agricultural ecological system.

## Keywords

Impulsive effects Delays Permanence Almost periodic solution Asymptotical stability## 1 Introduction

It is well known that there are two kinds of controlling strategies in the agricultural pest management: biological control and chemical control. The biological control is utilizing the predation of natural enemies to control the number of pests, while the chemical control is to achieve the rapid reduction of pests by artificial insecticides. Naturally, the enemies of the pests and the crops will also be affected and decrease in the process of artificial insecticides more or less; see [1].

As the dynamics in the agricultural ecological system is concerned, the specie of crops, the pests and the enemies constitute a food-chain system, and the artificial insecticides irregularly decreasing the number of species can be described by impulsive perturbations on the food-chain system; see [1, 2, 3, 4] etc.

In addition, it is reported that when the predators have to search, share or compete for food, a predator-dependent functional response is more reasonable in many situations. And there is much significant evidence in a laboratory and natural systems. It has been proved that models with a ratio-dependent functional response can exhibit much richer, more complicated and more reasonable or acceptable dynamics since Arditi and Ginzburg proposed the ratio-dependent predator–prey model in [8], which attracted the interest of many scholars; see [9, 10, 11, 12, 13].

*t*, respectively. \(r_{i}(t)\) and \(d_{i}(t)\) denote the intrinsic growth rate and the inner density resistance of them. \(a_{i}(t)\), \(b_{i}(t)\), \(m_{ij}(t)\) are the coefficients functions of the functional response. \(t_{k}\) is the impulsive controlling time, \(\tau _{1},\tau _{2}>0\) is the digest delay and \(q_{ik}>-1\) is the impulsive controlling constants, where \(q_{ik}>0\) means planting and \(q_{ik}<0\) means harvest. For an agricultural ecological control system, the absolute value of the \(q_{ik}\) is meant to acknowledge the poisonousness of the pesticide when \(q_{ik}<0\), \(i,j=1\), \(2,3; k\in N\).

- (H1)
All the function mentioned above such as \(r_{i}(t)\), \(d _{i}(t)\), \(r_{i}(t)\), \(a_{i}(t)\), \(b_{i}(t)\), \(m_{ij}(t)\) (\(i,j=1,2,3\)) are all bounded and positive almost periodic functions;

- (H2)
\(Q_{i}(t)= \prod_{0< t_{k}< t}(1+q_{ik})\) is almost period functions and there exist positive constants \(Q^{L}_{i}\) and \(Q^{M}_{i}\) such that \(Q^{L}_{i} \leq Q_{i}(t)\leq Q^{M}_{i}\), \(i=1,2,3\).

In the next section, we will give some useful lemmas and then prove our main results such as permanence of the system, and the existence and the uniqueness of an almost periodic solution which is uniformly asymptotically stable by constructing a series of Lyapunov functionals. In the last section, we give some numerical examples to support our theoretical results, then we provide a brief discussion and a summary of our main results.

## 2 Preliminaries

Denote \(K= \{{t_{k}}\in R |t_{k}< t_{k+1}, \lim_{k\rightarrow \pm \infty }t_{k}=\pm \infty , k\in N \}\), in which all the sets of all sequences are unbounded and increasing. Let \(\varOmega \subset R\), \(\varOmega \neq \varPhi \), \(\tau = \max_{1\leq i\leq 2}\{\tau _{i}\}\), \(\xi _{0}\in R\). Also, we denote by \(PC(\xi _{0})\) the space of all functions \(\phi :[\xi _{0}-\tau ,\xi _{0}]\rightarrow \varOmega \) having points of discontinuity at \(\mu _{1},\mu _{2},\ldots \in [\xi _{0}-\tau ,\xi _{0}]\) of the first kind and being left continuous at these points.

For \(J\subset R, PC(J,R)\) is the space of all piecewise continuous functions from *J* to *R* with points of discontinuity of the first kind \(t_{k}\), at which it is left continuous.

Then we have the following lemmas.

### Lemma 2.1

*Assume that*\((u(t),v(t),w(t))^{T}\)*is any solution of system* (5) *with initial conditions* (6), *then*\(u(t)>0\), \(v(t)>0\), \(w(t)>0\)*for all*\(t\in R^{+}\).

### Proof

This completes the proof of this lemma. □

### Lemma 2.2

### Proof

For \((u(t),v(t),w(t))^{T}\) being a solution of system (5), the conclusion follows.

Note that \(u(t)\), \(v(t)\), \(w(t)\) are continuous on each interval \((t_{k}, t_{k+1}]\), in the following we only need to prove the continuity of them at the impulsive points \(t=t_{k}\).

Similarly, \(v(t^{+}_{k})=v(t^{-}_{k})=v(t_{k})\), \(w(t^{+}_{k})=w(t^{-} _{k})=w(t_{k})\). These means \(u(t)\), \(v(t)\), \(w(t)\) are continuous at the impulsive points \(t=t_{k}\).

Therefore, \(u(t)\), \(v(t)\), \(w(t)\) are continuous on the whole interval \([0,\infty )\). Then we complete the proof of the first conclusion.

Similarly, we can easily prove that the second conclusion also holds by the previous definitions of the function \(D_{i}(t)\) (\(i=1,2,3\)), \(B_{1}(t)\), \(B_{23}(t)\), \(M_{12}(t)\), \(M_{21}(t)\), \(M_{23}(t)\), \(M_{32}(t) \). □

### Definition 2.1

(see [15])

*t*uniformly for \(\varphi \in C_{B}\), \(\forall \rho >0\), \(\exists M(\rho )>0\) such that \(|f(t,\varphi )| \leq M(\rho )\) as \(t\in R\), \(\varphi \in C_{\rho }\), while \(x_{t}\in C _{B}\) is defined as \(x_{t}(s)=x(t+s)\) for \(s\in [-\tau ,0] \). Here

*associate product system*of (13).

By the conclusions of [14, 15], we have Lemma 2.3.

### Lemma 2.3

(see [15])

*For*\(\phi ,\psi \in C_{B}\),

*suppose that there exists a Lyapunov function*\(V(t,\phi ,\psi )\)

*defined on*\(R^{+}\times C_{B}\times C_{B}\)

*satisfying the following three conditions*:

- (1)
\(u (\|\phi -\psi \| ) \leq V(t,\phi ,\psi ) \leq v (\|\phi -\psi \| )\),

*where*\(u,v\in \mathcal{P}\) = {\(u:R ^{+}\rightarrow R^{+}| u\)*is continuous increasing function and*\(u(s)\rightarrow 0\),*as*\(s\rightarrow 0\)}; - (2)
*there exists a positive constant*\(L>0\),*such that for any*\(\bar{\phi },\bar{\psi }, \hat{\phi },\hat{\psi } \in C_{B}\)$$ \bigl\vert V(t,\bar{\phi },\bar{\psi })-V(t,\hat{\phi }, \hat{\psi )} \bigr\vert \leq L \bigl( \Vert \bar{\phi }-\hat{\phi } \Vert + \Vert \bar{\psi }-\hat{\psi } \Vert \bigr); $$ - (3)
\(D^{+}V(t,\phi ,\psi )|_{\text{(14)}}\leq -\gamma V(t,\phi ,\psi )\),

*where**γ**is a positive constant*.

*Further*, *assume that* (13) *has a solution*\(x(t,v,\phi )\)*such that*\(|x(t,v,\phi )|\leq B_{1}\)*for*\(t\geq v \geq 0\), \(B>B_{1}>0\). *Then system* (13) *has a unique almost periodic solution which is uniformly asymptotically stable*.

### Lemma 2.4

*For*\(a>0\), \(b>0\), \(u(0)=u _{0}>0\):

- (1)
*If the inequality*\(u'(t) \leq u(t) (a-bu(t) )\)*holds*,*then*\(\limsup_{t\rightarrow +\infty } u(t) \leq \frac{a}{b}\). - (2)
*If the inequality*\(u'(t) \geq u(t) (a-bu(t) )\)*holds*,*then*\(\liminf_{t\rightarrow +\infty }u(t) \geq \frac{a}{b}\).

## 3 Main results

- (H3)
\(r^{L}_{1}a^{L}_{1}>m^{M}_{12}Q^{M}_{2}v^{*}\);

- (H4)
\(r^{L}_{2}b^{L}_{2}>m^{M}_{23}\);

Then we have Theorem 3.1.

### Theorem 3.1

*Assume that the coefficients of system*(5)

*satisfy the conditions*(H1)

*–*(H4),

*then any positive solution*\((u(t),v(t),w(t))^{T}\)

*of system*(5)

*satisfies*

### Proof

Thus, combining (18) with (22), we complete the proof of this theorem. □

### Theorem 3.2

*Assume that*(H1)

*–*(H4)

*hold*,

*then any positive solution*\((x(t),y(t),z(t))^{T}\)

*of system*(2)

*satisfies*

*Here*\(M_{1}=Q^{M}_{1}u^{*}\), \(M_{2}=Q^{M}_{2}v^{*}\), \(M_{3}=Q^{M}_{3}w ^{*}; m_{1}=Q^{L}_{1}u_{*}\), \(m_{2}=Q^{L}_{2}v_{*}\), \(m_{3}=Q^{L}_{3}w_{*}\).

### Proof

This completes the proof of this theorem. □

### Remark

System (2) is permanent under the conditions (H1)–(H4).

Considering the ecological meanings, one of the most important problems which one is usually concerned with is: does the system have an almost periodic solution? The we wonder: if there exists an almost periodic solution, is it uniformly asymptotically stable or not?

### Theorem 3.3

*Assume that*(H1)

*–*(H4)

*hold*,

*furthermore assume that there exist three positive*\(\lambda _{1}\), \(\lambda _{2}\)

*and*\(\lambda _{3}\)

*satisfying*:

- (H5)
\(\lambda _{1}K_{1}>\lambda _{1}K_{2}+\lambda _{2}L_{4}\),

- (H6)
\(\lambda _{1}K_{3}+\lambda _{2}L_{1}>\lambda _{2}L_{2}+ \lambda _{3}N_{2}\),

*then there exists an unique almost periodic solution which is uniformly asymptotically stable for the almost periodic system*(2).

### Proof

First, we prove that system (5) has a unique uniformly asymptotically stable almost periodic solution.

*M*large enough such that

Let \(u,v\in C(R^{+},R^{+})\), choose \(u=\underline{\lambda }s\), \(v=\bar{ \lambda }s\), then the first condition of Lemma 2.3 is satisfied.

This means the second condition of Lemma 2.3 is satisfied.

Now, we only need to check the last condition of Lemma 2.3.

If we denote \(\delta =\min \{K_{1}-K_{2}-\frac{\lambda _{2}L_{4}}{ \lambda _{1}}, \frac{\lambda _{1}K_{3}}{\lambda _{2}}+L_{1}-L_{2}-\frac{ \lambda _{3}N_{2}}{\lambda _{2}}, \frac{\lambda _{2}L_{3}}{\lambda _{3}}+N _{1} \}\), then \(\delta >0\) by the inequalities (H5) and (H6) in this theorem.

This means the last condition of Lemma 2.3 is satisfied. Thus, by the lemma, system (31) admits a unique uniformly asymptotically stable almost periodic solution \((x(t),y(t),z(t))^{T}\). \((u(t),v(t),w(t))^{T}=(e ^{x(t)},e^{y(t)},e^{z(t)})^{T}\).

Moreover, by the transformation (30), we can conclude that system (5) admits a unique uniformly asymptotically stable almost periodic solution.

Finally, we will explain that system (2) has a unique uniformly asymptotically stable almost periodic solution.

Therefore, \((x(t),y(t),z(t))^{T}\) is the unique uniformly asymptotically stable almost periodic solution of system (2) because of the uniqueness and the uniformly asymptotical stability of \((u(t),v(t),w(t))^{T}\). This completes the proof of this theorem. □

## 4 Numerical simulations and discussions

In this section, we will give a numerical example to illustrate the feasibility of our analytical results, then some discussions of the effects of impulsive perturbations and time delays on the system are presented in the end of the paper.

Obviously, they are all positive, bounded and almost periodic functions, which satisfy the condition (H1) in the paper.

Similarly, we can check that conditions (H5) and (H6) in Theorem 3.3 are also satisfied for the above coefficients. By the conclusion of this theorem, system (2) admits a unique uniformly asymptotically stable almost periodic solution. In order to verify this point, we choose two different initial conditions as follows:

*Case* 1: \(\phi _{1}(s)=x_{1}(0)=1.05\), \(\phi _{2}(s)=y_{1}(0)=4.5\), \(\phi _{3}(s)=z_{1}(0)=1.4\), \(s \in [-\tau ,0]\);

*Case*2: \(\phi _{1}(s)=x_{2}(0)=1.2\), \(\phi _{2}(s)=y_{2}(0)=5.5\), \(\phi _{3}(s)=z_{2}(0)=1.6\), \(s \in [-\tau ,0]\), where the other coefficients are the same as above. And we solve system (2) numerically and plot the time-series of the each specie in these two cases in Fig. 2. On the one hand, we can observe the periodicity of the each population clearly by this figure. On the other hand, we can also conclude that, although the initial number of the three populations is different, the population density of each population becomes the same eventually over time. This means that there exists an almost periodic solution which is asymptotically stable.

Finally, we will discuss the effect of chemical control; for example, people often use pesticide on pest control. And we choose the following three different chemical control strengths for system (2):

*Case* a: \(q_{1k}=-0.05\), \(q_{2k}=-0.08\), \(q_{3k}=-0.04\), \(t_{k}=k\), \(t \in [0,50]\),

*Case* b: \(q_{1k}=-0.10\), \(q_{2k}=-0.20\), \(q_{3k}=-0.08\), \(t_{k}=k\), \(t \in [0,50]\),

*Case* c: \(q_{1k}=-0.20\), \(q_{2k}=-0.30\), \(q_{3k}=-0.16\), \(t_{k}=k\), \(t \in [0,50]\). with the same initial condition \(\phi _{1}(s)=x_{1}(0)=1.05\), \(\phi _{2}(s)=y_{1}(0)=4.5\), \(\phi _{3}(s)=z_{1}(0)=1.4\), \(s \in [-\tau ,0]\).

By the theoretical analysis and numerical simulations in this paper, we can conclude that appropriate chemical control and biological control strategies can guarantee the crops, pests and natural enemies in the agricultural ecological system to coexist in a certain range of quantities, and even control their quantity as a good cyclic behavior. On the contrary, excessive chemical control, such as only increasing the concentration of pesticides, may increase the resistance of pests and the pest population density will not decrease but increase, which leads to the ultimate failure of pest control.

## Notes

### Acknowledgements

The authors would like to express their deep gratitude to the editor and the anonymous referee for his/her careful reading and valuable comments.

### Availability of data and materials

The data set supporting the conclusions of this article is included within the article.

### Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors have read and approved the final paper.

### Funding

This work is supported by Sichuan Science and Technology Program under Grant 2017JY0336 and Hunan Science and Technology Program under Grant 2019JJ50399, Longshan Talent Research Fund of Southwest University of Science and Technology under Grant 17LZX670 and 18LZX622.

### Competing interests

The authors declare that they have no competing interests.

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