Oscillatory dynamics in a discrete predator-prey model with distributed delays

Abstract

This work aims to discuss a predator-prey system with distributed delay. Various conditions are presented to ensure the existence and global asymptotic stability of positive periodic solution of the involved model. The method is based on coincidence degree theory and the idea of Lyapunov function. At last, simulation results are presented to show the correctness of theoretical findings.

Introduction

It is well known that the qualitative analysis of predator-prey models is an interesting mathematical problem and has received great attention from both theoretical and mathematical biologists [1–5]. In particular, the periodic solutions are of great interest. During the past decades, a great deal of excellent results have been reported for a lot of different continuous or impulsive predator-prey models. For example, Zhang and Hou [6] investigated the four positive periodic solutions of a ratio-dependent predator-prey system with multiple exploited (or harvesting) terms. Liu and Yan [7] considered positive periodic solutions for a neutral delay ratio-dependent predator-prey model with a Holling type II functional response. Liu [8] dealt with the impulsive periodic oscillation of a predator-prey model with Hassell-Varley-Holling functional response. For more related work, one can see [9–27]. Dunkel [28] pointed out that feedback control item in predator-prey models depends on the population number for certain time past and also depends on the average of the population number for a period of time past. In particular, time delay often occur in predator-prey models due to the impact of all the past life history of the predators and preys on their present birth rates. In many cases, the time delay will extend over the entire past due to the intra-species and inter-species competition. Then there is a distribution of delays over a period of time, thus the distributed delays should be incorporated in predator-prey models.

The functional response plays a key role in characterizing the interaction of predators and preys. Based on the experiments of different kinds of species, Holling [29] proposed three types of functional responses: (I) f₁(u) = au, (II) $f_2\left(u\right)=\frac{au}{c+u},$ (III) $f_3\left(u\right)=\frac{a{u}^2}{c+u^2},$ where u(t) represents the prey density at time t, c > 0 is the half-saturation constant, a > 0 denotes the search rate of the predator. Holling type II functional response is most typical of predators that specialized on one or a few prey [29–33]. So in this paper, Holling type II functional response is introduced in model (1).

Motivated by the viewpoint, we proposed the following predator-prey model with Holling II functional response and distributed delays

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle \frac{d{x}_1}{dt}=x_1\left(t\right)[r_1\left(t\right)-{\int }_{-\infty }^t{k}_1(s-t)x_1\left(s\right)ds]}\hfill \\ {\displaystyle -\frac{x_1\left(t\right)}{1+m{x}_1\left(t\right)}{\int }_{-\infty }^t{k}_2(s-t)x_2\left(s\right)ds,}\hfill \\ {\displaystyle \frac{d{x}_2}{dt}=x_2\left(t\right)[-r_2\left(t\right)-{\int }_{-\infty }^t{k}_3(s-t)x_2\left(s\right)ds]}\hfill \\ {\displaystyle +x_2\left(t\right){\int }_{-\infty }^t\frac{k_4(s-t)x_1\left(s\right)}{1+m{x}_1\left(t\right)}ds,}\hfill \end{array}\end{array}$$ (1)

where x_i(t)(i = 1, 2) stands for the prey and predator density at time t, r₁(t) denotes the intrinsic growth rate of prey at time t and r₂(t) denotes the death rate of predator at time t, m > 0 stands for the half-saturation constant, k_i: (−∞, 0] → (0, +∞)(i = 1, 2, 3, 4) is continuous function such that ${\int }_{-\infty }^t{k}_i\left(s\right)ds=1,{\int }_0^{\infty }s{k}_i\left(s\right)ds<\infty .$ For the biological meaning of model (1), one can see [34].

As pointed out in [35–42], discrete time models are more better to describe the dynamical behaviors than continuous ones since the populations have non-overlapping generations. What’s more, discrete-time systems can provide convenience for numerical simulations. Thus it is interesting to investigate discrete-time systems. The principle aim of this paper is to propose a discrete version of system (1) and analyze the effect of the periodicity of the ecological and environmental parameters on the dynamics of discrete time predator-prey model.

Discrete version of system (1)

Following [40, 43] and assuming that the average growth rates in system (1) change at regular intervals of time, one has

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle \frac{1}{x_1\left(t\right)}{\dot{x}}_1\left(t\right)=r_1\left(\left[t\right]\right)-\sum _{l=0}^{+\infty }{k}_1(-l)x_1(\left[t\right]-l)-\frac{1}{1+m{x}_1\left(\left[t\right]\right)}\sum _{l=0}^{+\infty }{k}_2(-l)x_2(\left[t\right]-l),}\hfill \\ {\displaystyle \frac{1}{x_2\left(t\right)}{\dot{x}}_2\left(t\right)=-r_2\left(\left[t\right]\right)-\sum _{l=0}^{+\infty }{k}_3(-l)x_2(\left[t\right]-l)+\sum _{l=0}^{+\infty }\frac{k_4(-l)x_1(\left[t\right]-l)}{1+m{x}_1\left(\left[t\right]\right)},}\hfill \end{array}\end{array}$$ (2)

where [t] stands for the integer part of t, t ∈ (0, +∞) and t ≠ 0, 1, 2, ⋯. The solution $\overline{x}={(x_1,x_2)}^T$ of (2) possesses the following natures:

1. $\overline{x}$ is continuous on [0, +∞).

2. $\frac{d{x}_1\left(t\right)}{dt},\frac{d{x}_2\left(t\right)}{dt}$ exist for ∀ t ∈ [0, +∞) with the possible exception of the points t ∈ {0, 1, 2, ⋯}, where left-sided derivative exists.

3. (2) holds ∀ [k, k + 1), where k = 0, 1, 2, ⋯.

Integrating (2) on [k, k + 1), k = 0, 1, 2, ⋯, one has

$$\begin{array}{c}\hfill \{\begin{array}{c}{x}_1\left(t\right)=x_1\left(k\right)exp\left\{\right[{\displaystyle r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l)x_1(k-l)}\hfill \\ {\displaystyle {\displaystyle -\frac{1}{1+m{x}_1\left(k\right)}\sum _{l=0}^{+\infty }{k}_2(-l)x_2(k-l)}\left](t-k)\right\},}\hfill \\ x_2\left(t\right)=x_2\left(k\right)exp\left\{\right[{\displaystyle -r_2\left(k\right)-\sum _{l=0}^{+\infty }{k}_3(-l)x_2(k-l)}\hfill \\ {\displaystyle {\displaystyle +\sum _{l=0}^{+\infty }\frac{k_4(-l)x_1(k-l)}{1+m{x}_1\left(k\right)}}\left](t-k)\right\}.}\hfill \end{array}\end{array}$$ (3)

Let t → k + 1, then (3) reads as

$$\begin{array}{c}\hfill \{\begin{array}{c}{x}_1(k+1)=x_1\left(k\right)exp\left\{\right[{\displaystyle r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l)x_1(k-l)}\hfill \\ {\displaystyle {\displaystyle -\frac{1}{1+m{x}_1\left(k\right)}\sum _{l=0}^{+\infty }{k}_2(-l)x_2(k-l)}\left]\right\},}\hfill \\ x_2(k+1)=x_2\left(k\right)exp\left\{\right[{\displaystyle -r_2\left(k\right)-\sum _{l=0}^{+\infty }{k}_3(-l)x_2(k-l)}\hfill \\ {\displaystyle {\displaystyle +\sum _{l=0}^{+\infty }\frac{k_4(-l)x_1(k-l)}{1+m{x}_1\left(k\right)}}\left]\right\},}\hfill \end{array}\end{array}$$ (4)

which is a discrete version of system (1), where k = 0, 1, 2, ⋯.

The following assumptions are made:

(H1) r_i: Z → R⁺ is positive α-periodic (α is a positive integer), i.e., r_i(k + α) = r_i(k)(i = 1, 2), ∀ k ∈ Z.

(H2) The following inequalities hold true.

$$\begin{array}{ccc}& & {\displaystyle 0\le \sum _{l=0}^{+\infty }{k}_1(-l)<+\infty ,0\le \sum _{l=0}^{+\infty }{k}_2(-l)<+\infty ,}\hfill \\ & & {\displaystyle 0\le \sum _{l=0}^{+\infty }{k}_3(-l)<+\infty ,0\le \sum _{l=0}^{+\infty }{k}_4(-l)<+\infty .}\hfill \end{array}$$

Existence of positive periodic solutions

First we given two notations:

$$\begin{array}{c}\hfill I_{\alpha }≔\{0,1,2,\cdots ,\alpha -1\},\overline{\ell }≔\frac{1}{\alpha }\sum _{k=0}^{\alpha -1}\ell \left(k\right),\end{array}$$

where ℓ(k) is a α–periodic sequence of real numbers defined for k ∈ Z. Let X, Y be normed vector spaces, L: DomL ⊂ X → Y be a linear mapping, N: X → Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL = codimImL < +∞ and ImL be closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projectors P: X → X and Q: Y → Y such that ImP = KerL, ImL = KerQ = Im(I − Q), it follows that L∣DomL ∩ KerP: (I − P)X → ImL is invertible. We denote the inverse of this map by K_P. If Ω is an open bounded subset of X, the mapping N will be called L–compact on $\overline{\Omega }$ if $QN\left(\overline{\Omega }\right)$ is bounded and $K_P(I-Q)N:\overline{\Omega }\to X$ is compact. Since ImQ is isomorphic to KerL, there exists a isomorphism J: ImQ → KerL.

Lemma 1 [44] Let L be a Fredholm mapping of index zero and let N be L–compact on $\overline{\Omega }.$ If

(a) ∀ ρ ∈ (0, 1), every solution y of Ly = ρNy is such that y ∉ ∂Ω;

(b) QNy ≠ 0, ∀ x ∈ KerL ⋂ ∂Ω, and deg{JQN, Ω ⋂ KerL, 0} ≠ 0, then the equation Ly = Ny has at least one solution lying in $DomL\cap \overline{\Omega }.$

Lemma 2 [40] Let h: Z → R be α periodic, i.e., h(k + α) = h(k), then ∀ ς₁, ς₂ ∈ I_α and ∀ k ∈ Z, one has

$$\begin{array}{c}\hfill h\left(k\right)\le h\left({\varsigma }_1\right)+\sum _{s=0}^{\alpha -1}|h(s+1)-g\left(s\right)|,\\ \hfill h\left(k\right)\ge h\left({\varsigma }_2\right)-\sum _{s=0}^{\alpha -1}|h(s+1)-g\left(s\right)|.\end{array}$$

Lemma 3 $({\widehat{x}}_1\left(k\right),{\widehat{x}}_2\left(k\right))$ is an α periodic solution of (4) with strictly positive components if and only if $(ln\left\{{\widehat{x}}_1\left(k\right)\right\},ln\left\{{\widehat{x}}_2\left(k\right)\right\}\})$ is an α periodic solution of

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle x_1(k+1)-x_1\left(k\right)=r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1(k-l)\right)}\hfill \\ {\displaystyle -\frac{1}{1+mexp\left(x_1\left(k\right)\right)}\sum _{l=0}^{+\infty }{k}_2(-l)exp\left(x_2(k-l)\right),}\hfill \\ {\displaystyle x_2(k+1)-x_2\left(k\right)=-r_2\left(k\right)-\sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2(k-l)\right)}\hfill \\ {\displaystyle +\sum _{l=0}^{+\infty }\frac{k_4(-l)exp\left(x_1(k-l)\right)}{1+mexp\left(x_1\left(k\right)\right)}.}\hfill \end{array}\end{array}$$ (5)

Proof If $({\widehat{x}}_1\left(k\right),{\widehat{x}}_2\left(k\right))$ is an α periodic solution of (4) with strictly positive components, then

$$\begin{array}{c}\hfill \{\begin{array}{c}{\widehat{x}}_1(k+1)={\widehat{x}}_1\left(k\right)exp\left\{\right[{\displaystyle r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l){\widehat{x}}_1(k-l)}\hfill \\ {\displaystyle {\displaystyle -\frac{1}{1+m{\widehat{x}}_1\left(k\right)}\sum _{l=0}^{+\infty }{k}_2(-l){\widehat{x}}_2(k-l)}\left]\right\},}\hfill \\ {\widehat{x}}_2(k+1)={\widehat{x}}_2\left(k\right)exp\left\{\right[{\displaystyle -r_2\left(k\right)-\sum _{l=0}^{+\infty }{k}_3(-l){\widehat{x}}_2(k-l)}\hfill \\ {\displaystyle {\displaystyle +\sum _{l=0}^{+\infty }\frac{k_4(-l){\widehat{x}}_1(k-l)}{1+m{\widehat{x}}_1\left(k\right)}}\left]\right\}.}\hfill \end{array}\end{array}$$

Hence

$$\begin{array}{c}\hfill \{\begin{array}{c}ln{\widehat{x}}_1(k+1)=ln\{{\widehat{x}}_1\left(k\right)exp\{[{\displaystyle r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l){\widehat{x}}_1(k-l)}\hfill \\ {\displaystyle {\displaystyle -\frac{1}{1+m{\widehat{x}}_1\left(k\right)}\sum _{l=0}^{+\infty }{k}_2(-l){\widehat{x}}_2(k-l)}\left]\right\}\},}\hfill \\ \text{ln}{\widehat{x}}_2(k+1)=\text{ln}\{{\widehat{x}}_2\left(k\right)exp\{[{\displaystyle -r_2\left(k\right)-\sum _{l=0}^{+\infty }{k}_3(-l){\widehat{x}}_2(k-l)}\hfill \\ {\displaystyle {\displaystyle +\sum _{l=0}^{+\infty }\frac{k_4(-l){\widehat{x}}_1(k-l)}{1+m{\widehat{x}}_1\left(k\right)}}\left]\right\}\},}\hfill \end{array}\end{array}$$

which leads to (5). If $(ln\left\{{\widehat{x}}_1\left(k\right)\right\},ln\left\{{\widehat{x}}_2\left(k\right)\right\}\})$ is an α periodic solution of (5), then

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle ln\left\{{\widehat{x}}_1(k+1)\right\}-ln\left\{{\widehat{x}}_1\left(k\right)\right\}=r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l)exp(ln\left\{{\widehat{x}}_1(k-l)\right\})}\hfill \\ {\displaystyle -\frac{1}{1+mexp(ln\left\{{\widehat{x}}_1\left(k\right)\right\})}\sum _{l=0}^{+\infty }{k}_2(-l)exp(ln\left\{{\widehat{x}}_2(k-l)\right\}),}\hfill \\ {\displaystyle ln\left\{{\widehat{x}}_2(k+1)\right\}-ln\left\{{\widehat{x}}_2\left(k\right)\right\}=-r_2\left(k\right)-\sum _{l=0}^{+\infty }{k}_3(-l)exp(ln\left\{{\widehat{x}}_2(k-l)\right\})}\hfill \\ {\displaystyle +\sum _{l=0}^{+\infty }\frac{k_4(-l)exp(ln\left\{{\widehat{x}}_1(k-l)\right\})}{1+mexp(ln\left\{{\widehat{x}}_1\left(k\right)\right\})},}\hfill \end{array}\end{array}$$

which leads to (4).

Define

$$\begin{array}{c}\hfill l_2=\{v=\left\{v\left(k\right)\right\}:v\left(k\right)\in R^2,k\in Z\}.\end{array}$$

Define |ζ| = max{|ζ₁|, |ζ₂|}, where ζ = (ζ₁, ζ₂)^T ∈ R². Let l^α ⊂ l₂ denote the subspace of all α periodic sequences equipped with the norm $\parallel v\parallel ={max}_{k\in I_{\omega }}\left|v\left(k\right)\right|,$ ∀ v = {v(k):k ∈ Z} ∈ l^ω. Then l_ω is a finite-dimensional Banach space.

Let

$$\begin{array}{c}\hfill l_0^{\alpha }=\{v=\left\{v\left(k\right)\right\}\in l^{\alpha }:\sum _{k=0}^{\alpha -1}v\left(k\right)=0\},\end{array}$$ (6)

$$\begin{array}{c}\hfill l_c^{\alpha }=\{v=\left\{v\left(k\right)\right\}\in l^{\alpha }:v\left(k\right)=h\in R^2,k\in Z\},\end{array}$$ (7)

then it follows that $l_0^{\alpha }$ and $l_c^{\alpha }$ are both closed linear subspaces of l^α and

$$\begin{array}{c}\hfill l^{\alpha }=l_0^{\alpha }+l_c^{\alpha },\text{dim}{l}_c^{\alpha }=2.\end{array}$$

Theorem 1 Let χ₉ be defined by (32). Suppose that (H1), (H2) and $\left(H3\right){\overline{r}}_1>{\sum }_{l=0}^{+\infty }{k}_2(-l)exp\left({\chi }_9\right)$ hold, then system (4) has at least an α periodic solution with positive components.

Proof. Let X = Y = l^α,

$$\left(Lv\right)\left(k\right)=v\left(k+1\right)-v\left(k\right)=\left[\begin{array}{c}{x}_1\left(k+1\right)-x_1\left(k\right)\\ x_2\left(k+1\right)-x_2\left(k\right)\end{array}\right],$$ (8)

$$\left(Nv\right)\left(k\right)=\left[\begin{array}{c}{f}_1\left(k\right)\\ f_2\left(k\right)\end{array}\right],$$ (9)

where v ∈ X, k ∈ Z and

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle f_1\left(k\right)=r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1(k-l)\right)}\hfill \\ {\displaystyle -\frac{1}{1+mexp\left(x_1\left(k\right)\right)}\sum _{l=0}^{+\infty }{k}_2(-l)exp\left(x_2(k-l)\right),}\hfill \\ {\displaystyle f_2\left(k\right)=-r_2\left(k\right)-\sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2(k-l)\right)}\hfill \\ {\displaystyle +\sum _{l=0}^{+\infty }\frac{k_4(-l)exp\left(x_1(k-l)\right)}{1+mexp\left(x_1\left(k\right)\right)}.}\hfill \end{array}\end{array}$$ (10)

Then L is a bounded linear operator and

$$\begin{array}{c}\hfill \text{Ker}L=l_c^{\alpha },\text{Im}L=l_0^{\alpha }\end{array}$$

and

$$\begin{array}{c}\hfill \text{dimKer}L=2=\text{codim}\text{Im}L,\end{array}$$

then L is a Fredholm mapping of index zero. Define

$$\begin{array}{c}\hfill Py=\frac{1}{\alpha }\sum _{s=0}^{\alpha -1}y\left(s\right),y\in X,Qz=\frac{1}{\alpha }\sum _{s=0}^{\omega -1}v\left(s\right),v\in Y.\end{array}$$

Then P and Q are continuous projectors such that

$$\begin{array}{c}\hfill \text{Im}P=\text{Ker}L,\text{Im}L=\text{Ker}Q=\text{Im}(I-Q).\end{array}$$

In addition, K_P: ImL → KerP ⋂ DomL exists and

$$\begin{array}{c}\hfill K_P\left(v\right)=\sum _{s=0}^{\alpha -1}v\left(s\right)-\frac{1}{\alpha }\sum _{s=0}^{\alpha -1}(\alpha -s)v\left(s\right).\end{array}$$

By the equation Lv = ρNv, ρ ∈ (0, 1), one gets

$$\begin{array}{c}\hfill \{\begin{array}{c}{x}_1(k+1)-x_1\left(k\right)=\rho [{\displaystyle r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1(k-l)\right)}\hfill \\ {\displaystyle {\displaystyle -\frac{1}{1+mexp\left(x_1\left(k\right)\right)}\sum _{l=0}^{+\infty }{k}_2(-l)exp\left(x_2(k-l)\right)}],}\hfill \\ x_2(k+1)-x_2\left(k\right)=\rho [{\displaystyle -r_2\left(k\right)-\sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2(k-l)\right)}\hfill \\ {\displaystyle {\displaystyle +\sum _{l=0}^{+\infty }\frac{k_4(-l)exp\left(x_1(k-l)\right)}{1+mexp\left(x_1\left(k\right)\right)}}].}\hfill \end{array}\end{array}$$ (11)

Suppose that v(k) = (x₁(k), x₂(k))^T ∈ X is an arbitrary solution of system (11) for a certain ρ ∈ (0, 1) then one has

$$\begin{array}{ccc}& & \sum _{k=0}^{\alpha -1}[{\displaystyle \sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1(k-l)\right)}\hfill \\ & & {\displaystyle +\frac{1}{1+mexp\left(x_1\left(k\right)\right)}\sum _{l=0}^{+\infty }{k}_2(-l)exp\left(x_2(k-l)\right)}]={\overline{r}}_1\alpha ,\hfill \end{array}$$ (12)

$$\begin{array}{ccc}& & \sum _{k=0}^{\alpha -1}\left[{\displaystyle \sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2(k-l)\right)-\sum _{l=0}^{+\infty }\frac{k_4(-l)exp\left(x_1(k-l)\right)}{1+mexp\left(x_1\left(k\right)\right)}}\right]={\overline{r}}_2\alpha .\hfill \end{array}$$ (13)

It follows from (11)–(13) that

$$\begin{array}{c}\hfill \sum _{k=0}^{\alpha -1}|x_1(k+1)-x_1\left(k\right)|\le 2{\overline{r}}_1\alpha ,\end{array}$$ (14)

$$\begin{array}{c}\hfill \sum _{k=0}^{\alpha -1}|x_2(k+1)-x_2\left(k\right)|\le 2{\overline{r}}_2\alpha .\end{array}$$ (15)

If v = {v(k)} ∈ X, then ∃ ξ_i, η_i ∈ I_α such that

$$\begin{array}{c}\hfill x_i\left({\xi }_i\right)=\underset{k\in I_{\alpha }}{min}\left\{x_i\left(k\right)\right\},x_i\left({\eta }_i\right)=\underset{k\in I_{\alpha }}{max}\left\{x_i\left(k\right)\right\}(i=1,2).\end{array}$$ (16)

By (12) and (13), we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1\left({\xi }_1\right)\right)\le \sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1(k-l)\right)<{\overline{r}}_1\alpha ,}\end{array}$$ (17)

$$\begin{array}{c}\hfill {\displaystyle \sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2\left({\eta }_2\right)\right)\ge \sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2(k-l)\right)>{\overline{r}}_2\alpha .}\end{array}$$ (18)

Thus

$$\begin{array}{c}\hfill x_1\left({\xi }_1\right)<ln\left[\frac{{\overline{r}}_1}{{\displaystyle \sum _{l=0}^{+\infty }{k}_1(-l)}}\right]≔{\theta }_1,\end{array}$$ (19)

$$\begin{array}{c}\hfill x_2\left({\eta }_2\right)>ln\left[\frac{{\overline{r}}_2}{{\displaystyle \sum _{l=0}^{+\infty }{k}_3(-l)}}\right]≔{\delta }_2.\end{array}$$ (20)

In the sequel, we consider two cases.

(a) If x₁(η₁) ≥ x₂(η₂), then it follows from (12) that

$$\begin{array}{c}\hfill \left[{\displaystyle \sum _{l=0}^{+\infty }{k}_1(-l)+\sum _{l=0}^{+\infty }{k}_2(-l)}\right]exp\left(x_1\left({\eta }_1\right)\right)\alpha \ge {\overline{r}}_1\alpha \end{array}$$

which leads to

$$\begin{array}{c}\hfill x_1\left({\eta }_1\right)>ln\left[\frac{{\overline{r}}_1}{{\displaystyle \sum _{l=0}^{+\infty }{k}_1(-l)+\sum _{l=0}^{+\infty }{k}_2(-l)}}\right]≔{\theta }_2,\end{array}$$ (21)

It follows from (19),(21) and Lemma 2 that

$$\begin{array}{ccc}\hfill x_1\left(k\right)& \le & x_1\left({\xi }_1\right)+\sum _{s=0}^{\alpha -1}|x_1(s+1)-x_1\left(s\right)|\hfill \\ & \le & {\theta }_1+2{\overline{r}}_1\alpha ≔{\chi }_1,\hfill \end{array}$$ (22)

$$\begin{array}{ccc}\hfill x_1\left(k\right)& \ge & x_1\left({\eta }_1\right)-\sum _{s=0}^{\alpha -1}|x_1(s+1)-x_1\left(s\right)|\hfill \\ & \ge & {\theta }_2-2{\overline{r}}_1\alpha ≔{\chi }_2.\hfill \end{array}$$ (23)

By (22) and (23), we derive

$$\begin{array}{c}\hfill \underset{k\in I_{\alpha }}{max}\left\{x_1\left(k\right)\right\}\le max\left\{\right|{\chi }_1|,|{\chi }_2\left|\right\}≔{\chi }_3.\end{array}$$ (24)

From (13) and (24), we obtain

$$\begin{array}{c}\hfill {\displaystyle \sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2\left({\xi }_2\right)\right)\alpha -\sum _{l=0}^{+\infty }{k}_4(-l)exp\left({\chi }_3\right)\alpha \le {\overline{r}}_2\alpha .}\end{array}$$

Then

$$\begin{array}{c}\hfill x_2\left({\xi }_2\right)\le ln\left[\frac{{\displaystyle {\overline{r}}_2+\sum _{l=0}^{+\infty }{k}_4(-l)exp\left({\chi }_3\right)}}{{\displaystyle \sum _{l=0}^{+\infty }{k}_3(-l)}}\right]≔{\delta }_1.\end{array}$$ (25)

Thus by (20), (25) and Lemma 3.2, we get

$$\begin{array}{ccc}\hfill x_2\left(k\right)& \le & x_2\left({\xi }_2\right)+\sum _{s=0}^{\alpha -1}|x_2(s+1)-x_2\left(s\right)|\hfill \\ & \le & {\delta }_1+2{\overline{r}}_2\alpha ≔{\chi }_4,\hfill \end{array}$$ (26)

$$\begin{array}{ccc}\hfill x_2\left(k\right)& \ge & x_2\left({\eta }_2\right)-\sum _{s=0}^{\alpha -1}|x_2(s+1)-x_2\left(s\right)|\hfill \\ & \ge & {\delta }_2-2{\overline{r}}_2\alpha ≔{\chi }_5.\hfill \end{array}$$ (27)

It follows from (26) and (27) that

$$\begin{array}{c}\hfill \underset{k\in I_{\alpha }}{max}\left\{x_2\left(k\right)\right\}\le max\left\{\right|{\chi }_4|,|{\chi }_5\left|\right\}≔{\chi }_6.\end{array}$$ (28)

(b) If x₁(η₁) < x₂(η₂), then it follows from (13) that

$$\begin{array}{c}\hfill {\displaystyle \sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2\left({\xi }_2\right)\right)\alpha -\frac{1}{m}\sum _{l=0}^{+\infty }{k}_4(-l)\alpha \le {\overline{r}}_2\alpha }\end{array}$$

which leads to

$$\begin{array}{c}\hfill x_2\left({\xi }_2\right)<ln\left[\frac{{\displaystyle {\overline{r}}_2+\frac{1}{m}\sum _{l=0}^{+\infty }{k}_4(-l)}}{{\displaystyle \sum _{l=0}^{+\infty }{k}_3(-l)}}\right]≔{\overline{\delta }}_1,\end{array}$$ (29)

It follows from (20),(29) and Lemma 3.2 that

$$\begin{array}{ccc}\hfill x_2\left(k\right)& \le & x_2\left({\xi }_2\right)+\sum _{s=0}^{\alpha -1}|x_2(s+1)-x_2\left(s\right)|\hfill \\ & \le & {\overline{\delta }}_1+2{\overline{r}}_2\alpha ≔{\chi }_7,\hfill \end{array}$$ (30)

$$\begin{array}{ccc}\hfill x_2\left(k\right)& \ge & x_2\left({\eta }_2\right)-\sum _{s=0}^{\alpha -1}|x_2(s+1)-x_2\left(s\right)|\hfill \\ & \ge & {\delta }_2-2{\overline{r}}_2\alpha ≔{\chi }_8.\hfill \end{array}$$ (31)

By (30) and (31), we derive

$$\begin{array}{c}\hfill \underset{k\in I_{\alpha }}{max}\left\{x_2\left(k\right)\right\}\le max\left\{\right|{\chi }_7|,|{\chi }_8\left|\right\}≔{\chi }_9.\end{array}$$ (32)

From (12), we get

$$\begin{array}{c}\hfill \left[{\displaystyle \sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1\left({\eta }_1\right)\right)\alpha +\sum _{l=0}^{+\infty }{k}_2(-l)exp\left({\chi }_9\right)\alpha }\right]\ge {\overline{r}}_1\alpha .\end{array}$$

Then

$$\begin{array}{c}\hfill x_1\left({\eta }_1\right)\ge ln\left[\frac{{\displaystyle {\overline{r}}_1-\sum _{l=0}^{+\infty }{k}_2(-l)exp\left({\chi }_9\right)}}{{\displaystyle \sum _{l=0}^{+\infty }{k}_1(-l)}}\right]≔{\overline{\theta }}_2.\end{array}$$ (33)

Thus by (19), (33) and Lemma 3.2, we get

$$\begin{array}{ccc}\hfill x_1\left(k\right)& \le & x_1\left({\xi }_1\right)+\sum _{s=0}^{\alpha -1}|x_1(s+1)-x_1\left(s\right)|\hfill \\ & \le & {\delta }_1+2{\overline{r}}_2\alpha ≔{\chi }_{10},\hfill \end{array}$$ (34)

$$\begin{array}{ccc}\hfill x_1\left(k\right)& \ge & x_1\left({\eta }_1\right)-\sum _{s=0}^{\alpha -1}|x_1(s+1)-x_1\left(s\right)|\hfill \\ & \ge & {\overline{\theta }}_2-2{\overline{r}}_2\alpha ≔{\chi }_{11}.\hfill \end{array}$$ (35)

It follows from (34) and (35) that

$$\begin{array}{c}\hfill \underset{k\in I_{\alpha }}{max}\left\{x_1\left(k\right)\right\}\le max\left\{\right|{\chi }_{10}|,|{\chi }_{11}\left|\right\}≔{\chi }_{12}.\end{array}$$ (36)

Then χ_i(i = 1, 2, ⋯, 11) has no relation with ρ ∈ (0, 1). Let M = max{χ₃, χ₆, χ₉, χ₁₂} + M₀, where M₀ > 0 which satisfies $max\left\{\right|ln\left\{x_1^{*}\right\}|,|ln\left\{x_2^{*}\right\}\left|\right\}<M_0$, where ${(x_1^{*},x_2^{*})}^T$ is the unique positive solution of (5). Thus any solution v = {v(k)} = {(x₁(k), x₂(k))^T} of (11) in X satisfies ‖v‖ < M, k ∈ Z.

Let Ω ≔ {v = {v(k)} ∈ X: ‖v‖ < M}, then Ω is an open, bounded set in X and (a) of Lemma 1 is satisfied. When v ∈ ∂Ω ∩ KerL, v = {(x₁, x₂)^T} with ‖v‖ = max{|x₁|, |x₂|} = M. Then

$$\begin{array}{c}\hfill QNz=\left[\begin{array}{c}{\Lambda }_1\\ {\Lambda }_2\end{array}\right]\ne 0,\end{array}$$

where

$$\begin{array}{ccc}& & {\displaystyle {\Lambda }_1={\overline{r}}_1-\sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1\right)-\sum _{k=0}^{\alpha -1}\frac{1}{1+mexp\left(x_1\right)}\sum _{l=0}^{+\infty }{k}_2(-l)exp\left(x_2\right),}\hfill \\ & & {\displaystyle {\Lambda }_2=-{\overline{r}}_2-\sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2\right)+\sum _{k=0}^{\alpha -1}\sum _{l=0}^{+\infty }\frac{k_4(-l)exp\left(x_1\right)}{1+mexp\left(x_1\right)}.}\hfill \end{array}$$

Let ϕ(x₁, x₂, μ) = μQNv + (1 − μ)Gv, μ ∈ [0, 1], where

$$\begin{array}{c}\hfill Gv=\left(\begin{array}{c}{\displaystyle {\overline{r}}_1-\sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1\right)}\\ {\displaystyle -{\overline{r}}_2-\sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2\right)}\end{array}\right).\end{array}$$

Letting J be the identity mapping, we have

$$\begin{array}{ccc}& & deg[JQN{(x_1,x_2)}^T;\Omega \cap \text{ker}L;0]\hfill \\ & & =deg[QN{(x_1,x_2)}^T;\Omega \cap \text{ker}L;0]\hfill \\ & & =deg\left[\varphi \right(x_1,x_2,1);\Omega \cap \text{ker}L;0]\hfill \\ & & =deg\left[\varphi \right(x_1,x_2,0);\Omega \cap \text{ker}L;0]\hfill \\ & & =\text{sign}\{det[\begin{array}{cc}{\displaystyle \sum _{l=0}^{+\infty }{k}_1(-l)exp\left(x_1^{*}\right)}& 0\\ 0& {\displaystyle -\sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2^{*}\right)}\end{array}\left]\right\}\hfill \\ & & =\text{sign}\left[{\displaystyle -\sum _{l=0}^{+\infty }{k}_1(-l)\sum _{l=0}^{+\infty }{k}_3(-l)exp\left(x_2^{*}\right)exp(x_1^{*}+x_2^{*})}\right]=-1\ne 0.\hfill \end{array}$$

It follows that Lv = Nv has at least one solution in $\text{Dom}L\cap \overline{\Omega }$, i.e., (5) has at least one α periodic solution in $\text{Dom}L\cap \overline{\Omega }$, say $v^{*}=\left\{v^{*}\left(k\right)\right\}=\left\{{(x_1^{*}\left(k\right),x_2^{*}\left(k\right))}^T\right\}$. Let ${\overline{x}}_1^{*}\left(k\right)=exp\left\{x_1^{*}\left(k\right)\right\},{\overline{x}}_2^{*}\left(k\right)=exp\left\{x_2^{*}\left(k\right)\right\}\}$ then by Lemma 3 we know that ${\overline{v}}^{*}=\left\{{\overline{x}}^{*}\left(k\right)\right\}={\{{\overline{x}}_1^{*}\left(k\right),{\overline{x}}_2^{*}\left(k\right))}^T\}$ is a α positive periodic solution of system (4). The proof is complete.

Global asymptotic stability

Let the delays be zero, then (4) becomes

$$\begin{array}{c}\hfill \{\begin{array}{c}{x}_1(k+1)=x_1\left(k\right)exp\left\{\right[r_1\left(k\right)-k_1{x}_1\left(k\right)-\frac{k_2{x}_2\left(k\right)}{1+m{x}_1\left(k\right)}\left]\right\},\hfill \\ x_2(k+1)=x_2\left(k\right)exp\left\{\right[-r_2\left(k\right)-k_3{x}_2\left(k\right)+\frac{k_4{x}_1\left(k\right)}{1+m{x}_1\left(k\right)}\left]\right\}.\hfill \end{array}\end{array}$$ (37)

Theorem 2 Assume that (H1) and (H2) are satisfied and furthermore suppose that there exist positive constants ν, σ₁ and σ₂ such that

$$\begin{array}{c}\hfill \{\begin{array}{c}{\sigma }_1[k_1+\frac{k_{2}m}{{(1+m{x}_1^{*})}^2}]-{\sigma }_2\left[\frac{1}{{(1+m{x}_1^{*})}^2}\right]>\nu ,\hfill \\ {\sigma }_2{k}_3-{\sigma }_1\left[\frac{m{x}_1^{*}\left(k\right)x_2^{*}\left(k\right)}{{(1+m{x}_1^{*})}^2}\right]>\nu .\hfill \end{array}\end{array}$$ (38)

Then the positive ω-periodic solution of system (37) is globally asymptotically stable.

Proof In view of Theorem 1, there exists a positive periodic solution $\{x_1^{*}\left(k\right),x_2^{*}\left(k\right)\}$ of system (37). Make the change of variable

$$\begin{array}{c}\hfill u_i\left(k\right)=x_i\left(k\right)-x_i^{*}\left(k\right)(i=1,2).\end{array}$$ (39)

It follows from (37) that

$$\begin{array}{ccc}\hfill u_1(k+1)& =& x_1(k+1)-x_1^{*}(k+1)\hfill \\ & =& x_1\left(k\right)exp\left\{\right[r_1\left(k\right)-k_1{x}_1\left(k\right)-\frac{k_2{x}_2\left(k\right)}{1+m{x}_1\left(k\right)}\left]\right\}\hfill \\ & & -x_1^{*}\left(k\right)exp\left\{\right[{\displaystyle r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l)x_1^{*}\left(k\right)-\frac{k_2{x}_2^{*}\left(k\right)}{1+m{x}_1^{*}\left(k\right)}}\left]\right\}\hfill \\ & =& \{x\left(k\right)exp[-(k_1+\frac{k_{2}m{x}_1^{*}\left(k\right)}{{(1+m{x}_1^{*})}^2})u_1\left(k\right)\hfill \\ & & -\left(\frac{k_2}{1+m{x}_1^{*}\left(k\right)}\right)u_2\left(k\right)]-x^{*}\left(k\right)\}\frac{x^{*}(k+1)}{x^{*}\left(k\right)}\hfill \\ & =& \left\{\right[1-(k_1{x}_1^{*}\left(k\right)+\frac{k_{2}m{x}_1^{*}\left(k\right)}{{(1+m{x}_1^{*})}^2})]\frac{u_1\left(k\right)}{x^{*}\left(k\right)}\hfill \\ & & -\left(\frac{m{x}_1^{*}\left(k\right)x_2^{*}\left(k\right)}{{(1+m{x}_1^{*})}^2}\right)u_2\left(k\right)+{\gamma }_1\}{x}^{*}(k+1),\hfill \end{array}$$ (40)

$$\begin{array}{ccc}\hfill u_2(k+1)& =& x_2(k+1)-x_2^{*}(k+1)\hfill \\ & =& x_2\left(k\right)exp\left\{\right[-r_2\left(k\right)-k_3{x}_2\left(k\right)+\frac{k_4{x}_1\left(k\right)}{1+m{x}_1\left(k\right)}\left]\right\}\hfill \\ & & -x_2^{*}\left(k\right)exp\left\{\right[-r_2\left(k\right)-k_3{x}_2^{*}\left(k\right)+\frac{k_4{x}_1^{*}\left(k\right)}{1+m{x}_1\left(k\right)}\left]\right\}\hfill \\ & =& \{x_2\left(k\right)exp[-k_3{u}_2\left(k\right)-(\frac{k_4{x}_1\left(k\right)}{1+m{x}_1\left(k\right)}-\frac{k_4{x}_1^{*}\left(k\right)}{1+m{x}_1\left(k\right)})]-x_2^{*}\left(k\right)\}\hfill \\ & & \times \frac{y^{*}(k+1)}{x_2^{*}\left(k\right)}\hfill \\ & =& \left\{\right[1-k_3{x}_2^{*}\left(k\right)]\frac{u_2\left(k\right)}{x_2^{*}\left(k\right)}-\frac{1}{{(1+m{x}_1^{*})}^2}{u}_1\left(k\right)+{\gamma }_2\}{x}_2^{*}(k+1).\hfill \end{array}$$ (41)

where $\frac{\left|\right|{\gamma }_i\left|\right|}{\left|\right|u\left|\right|}(i=1,2)$ converges to zero as ||u|| → 0.

Define a function V by

$$\begin{array}{c}\hfill V\left(N\left(k\right)\right)={\sigma }_1\left|\frac{u_1\left(k\right)}{x_1^{*}\left(k\right)}\right|+{\sigma }_2\left|\frac{u_2\left(k\right)}{x_2^{*}\left(k\right)}\right|,\end{array}$$ (42)

where σ₁ > 0 and σ₂ > 0 are given by (44) and (45) respectively. Calculating the difference of V along the solution of system (40) and (41), we have

$$\begin{array}{ccc}\hfill \Delta V& =& {\sigma }_1\left(\right|\frac{u_1(k+1)}{x_1^{*}(k+1)}-\frac{u_1\left(k\right)}{x_1^{*}\left(k\right)}\left|\right)+{\sigma }_2\left(\right|\frac{u_2(k+1)}{x_2^{*}(k+1)}-\frac{u_2\left(k\right)}{x_2^{*}\left(k\right)}\left|\right)\hfill \\ & \le & -{\sigma }_1[k_1+\frac{k_{2}m}{{(1+m{x}_1^{*})}^2}]|u_1\left(k\right)|+{\sigma }_1\left[\frac{m{x}_1^{*}\left(k\right)x_2^{*}\left(k\right)}{{(1+m{x}_1^{*})}^2}\right]\left|u_2\left(k\right)\right|\hfill \\ & & -{\sigma }_2{k}_3|u_2\left(k\right)|+{\sigma }_2\left[\frac{1}{{(1+m{x}_1^{*})}^2}\right]\left|u_1\left(k\right)\right|\hfill \\ & \le & -{\Pi }_1|u_1\left(k\right)|-{\Pi }_2\left|u_2\left(k\right)\right|,\hfill \end{array}$$ (43)

where

$${\Pi }_1={\sigma }_1\left[k_1+\frac{k_{2}m}{{\left(1+m{x}_1^{*}\right)}^2}\right]-{\sigma }_2\left[\frac{1}{{\left(1+m{x}_1^{*}\right)}^2}\right],$$ (44)

$${\Pi }_2={\sigma }_2{k}_3-{\sigma }_1\left[\frac{m{x}_1^{*}\left(k\right)x_2^{*}\left(k\right)}{{\left(1+m{x}_1^{*}\right)}^2}\right].$$ (45)

It follows from the condition (38) that ∃ ϵ > 0 such that, if k is sufficiently large and ||u|| < ϵ, then

$$\begin{array}{c}\hfill \Delta V\le -\frac{\nu }{2}\left\{\right|u_1\left(k\right)|+|u_2\left(k\right)\left|\right\}<-\frac{\nu ϵ}{2}.\end{array}$$ (46)

In view of Freedman [45], we can see that the trivial solutions of (40) and (41) is uniformly asymptotically stable and so is the solution {(x*(k), y*(k))^T} of (37). The proof is complete.

Remark 1 In [34], Ye et al. investigated the periodic solution of a continuous predator-prey system with Holling type II functional response and infinite delays by applying continuation theorem in coincidence degree theory and some priori estimates on solutions, moreover, this paper does not involve the global asymptotic stability. In this paper, we study the existence of periodic solution of discrete predator-prey model with distributed delays by applying continuation theorem in coincidence degree theory and analyze the global asymptotic stability of periodic solution by Lyapunov function. Form this viewpoint, the results of this article supplement the previous studies of Ye et al. [18].

Numerical example

Example 1 Consider the model as follows:

$$\begin{array}{c}\hfill \{\begin{array}{c}{x}_1(k+1)=x_1\left(k\right)exp\left\{\right[{\displaystyle r_1\left(k\right)-\sum _{l=0}^{+\infty }{k}_1(-l)x_1(k-l)}\hfill \\ {\displaystyle {\displaystyle -\frac{1}{1+m{x}_1\left(k\right)}\sum _{l=0}^{+\infty }{k}_2(-l)x_2(k-l)}\left]\right\},}\hfill \\ x_2(k+1)=x_2\left(k\right)exp\left\{\right[{\displaystyle -r_2\left(k\right)-\sum _{l=0}^{+\infty }{k}_3(-l)x_2(k-l)}\hfill \\ {\displaystyle {\displaystyle +\sum _{l=0}^{+\infty }\frac{k_4(-l)x_1(k-l)}{1+m{x}_1\left(k\right)}}\left]\right\},}\hfill \end{array}\end{array}$$ (47)

where r₁(k) = 0.6 + sin kπ, r₂(k) = 0.45 + sin kπ, m = 5, k_i(s) = e^s(i = 1, 2, 3, 4). So ${\overline{r}}_1=0.3,{\overline{r}}_2=0.225,{\sum }_{l=0}^{+\infty }{k}_2(-l)exp\left({\chi }_9\right)\approx 0.2437.$ Thus the conditions (H1)-(H3) of Theorem 3.1 hold true. Therefore, system (47) has at least a positive two-periodic solution (see Figs 1 and 2). Fig 1 shows the changing situation of prey density with the increase of time t; Fig 2 shows the changing situation of predator density with the increase of time t; From Figs 1 and 2, we can see that the prey density and the predator density will keep periodic oscillation with the increase of time t.

Fig 1. The time histories of t-x₁,t-x₂.

The blue line stands for x₁(t) and the red line stands for x₂(t).

Fig 2. The relational graph of t, x₁ and x₂.

Example 2 Consider the model as follows:

$$\begin{array}{c}\hfill \{\begin{array}{c}{x}_1(k+1)=x_1\left(k\right)exp\left\{\right[r_1\left(k\right)-k_1{x}_1\left(k\right)-\frac{k_2{x}_2\left(k\right)}{1+m{x}_1\left(k\right)}\left]\right\},\hfill \\ x_2(k+1)=x_2\left(k\right)exp\left\{\right[-r_2\left(k\right)-k_3{x}_2\left(k\right)+\frac{k_4{x}_1\left(k\right)}{1+m{x}_1\left(k\right)}\left]\right\}.\hfill \end{array}\end{array}$$ (48)

where r₁(k) = 0.2 + cos kπ, r₂(k) = 0.1 + cos kπ, m = 2, k₁ = 0.2, k₂ = 0.12, k₃ = 0.24, k₄ = 0.35,. So ${\overline{r}}_1=0.1,{\overline{r}}_2=0.05.$ Let σ₁ = 0.18, σ₂ = 0.23, ν = 0.04. Thus the conditions (H1)-(H2) and (38) of Theorem 4.1 are satisfied. Thus the positive two-periodic solution of system (48) is globally asymptotically stable (see Figs 3 and 4). Fig 3 shows the changing situation of prey density with the increase of time t; Fig 4 shows the changing situation of predator density with the increase of time t; From Figs 3 and 4, we can see that the prey density and the predator density will keep globally asymptotically stable periodic oscillation with the increase of time t.

Fig 3. The time histories of t-x₁,t-x₂.

The blue line stands for x₁(t) and the red line stands for x₂(t).

Fig 4. The relational graph of t, x₁ and x₂.

Conclusions

Based on the previous works and some biological meanings of predators and preys, we propose a new discrete delayed predator-prey system. By using the continuation theorem in coincidence degree theory, we present a set of sufficient conditions to ensure to ensure the existence of positive periodic solution of the discrete delayed predator-prey system. In addition, we also discussed the global asymptotic stability of positive periodic solution for the considered system. The obtained theoretical findings have important significance in biological ecology. Considering the effect of random factor, it is meaningful for us to deal with the dynamics of stochastic predator-prey system. This topic will be our future research direction.

Supporting information

S1 Fig. The time histories of t-x₁,t-x₂.

The blue line stands for x₁(t) and the red line stands for x₂(t).

(DOC)

S2 Fig. The relational graph of t, x₁ and x₂.

(DOC)

S3 Fig. The time histories of t-x₁,t-x₂.

The blue line stands for x₁(t) and the red line stands for x₂(t).

(DOC)

S4 Fig. The relational graph of t, x₁ and x₂.

(DOC)

Data Availability

All relevant data are within the manuscript and its Supporting Information file.

Funding Statement

This work is supported by National Natural Science Foundation of China (No.61673008) and Project of High-level Innovative Talents of Guizhou Province ([2016]5651), Major Research Project of The Innovation Group of The Education Department of Guizhou Province ([2017]039), Project of Key Laboratory of Guizhou Province with Financial and Physical Features ([2017]004) and Foundation of Science and Technology of Guizhou Province ([2018]1025 and [2018]1020).