Stability criterion of a nonautonomous 3-species ratio-dependent diffusive predator-prey model

Abstract

The global stability of a nonautonomous 3-species ratio-dependent diffusive predator-prey model is studied in this paper. Firstly, some easily verifiable sufficient conditions which guarantee the existence of the strictly positive space homogenous periodic solution (SHPS) of the ratio- dependent predator-prey model (RDPPM) with diffusive and variable coefficient are achieved by using a comparison theorem of differential equation and fixed point theorem. At the same time, some new analysis techniques are developed as a byproduct. Secondly, some sufficient conditions ensuring the globally asymptotically stability of the strictly positive SHPS of the diffusive nonautonomous predator-prey model are given by using the method of upper and lower solutions (UALS) for the parabolic partial differential equations and Lyapunov stability theory. In addition, two numerical examples are given to validate the theoretical results obtained in this paper.

1 Introduction

In this article, we focus on the following nonautonomous 3-species ratio-dependent diffusive predator-prey model

$$ \left \{ \textstyle\begin{array}{l} \partial u_{1}(x,t)/\partial t - d_{1}(t)\Delta u_{1}(x,t) \\ \quad = u_{1}(x,t)[r_{1}(t) - a_{11}(t)u_{1}(x,t) - a_{12}(t)u_{2}(x,t) - \frac{a_{13}(t)u_{3}(x,t)}{b_{13}(t)u_{3}(x,t) + u_{1}(x,t)}], \\ \partial u_{2}(x,t)/\partial t - d_{2}(t)\Delta u_{2}(x,t) \\ \quad = u_{2}(x,t)[r_{2}(t) - a_{22}(t)u_{2}(x,t) - a_{21}(t)u_{1}(x,t) - \frac{a_{23}(t)u_{3}(x,t)}{b_{23}(t)u_{3}(x,t) + u_{2}(x,t)}], \\ \partial u_{3}(x,t)/\partial t - d_{3}(t)\Delta u_{3}(x,t) \\ \quad = u_{3}(x,t)[ - r_{3}(t) + \frac{a_{31}(t)u_{1}(x,t)}{b_{13}(t)u_{3}(x,t) + u_{1}(x,t)} + \frac{a_{32}(t)u_{2}(x,t)}{b_{23}(t)u_{3}(x,t) + u_{2}(x,t)}], \end{array}\displaystyle \right . $$

(1.1)

with the Neumman boundary and initial conditions

$$ \partial u_{i}(x,t)/\partial n = 0, (x,t) \in \partial \Omega \times R^{ +}, u_{i}(x,0) = u_{i0}(x) > 0, x \in \Omega , i = 1,2,3, $$

(1.2)

here Ω is a bounded smooth domain in $R^{n}$ with boundary ∂Ω, Δ is a Laplace operator on Ω, $\partial /\partial n$ denotes the outward normal derivation on ∂Ω, $u_{i}(x,t)$ represent the density of i-th species at point $x = (x_{1}, \ldots ,x_{n})$ and the time of t. From Table 1, it can be seen that the biological significance of the parameters in model (1.1).

Table 1 The biological significance of parameters in model (1.1)

Based on the phenomenon that the birth rate, mortality rate, and interactions of the population vary periodically with the environment and the population size is not less than zero, so we assume in this article that the all coefficients of the diffusion nonautonomous RDPPM (1.1)–(1.2) are continuous and positive ω-periodic functions. Models (1.1)–(1.2) describe the interaction among 3-species and are an important model in biomathcmatics. For example, the relationship among eagles, weasels, and snakes. Eagles prey on both weasels and snakes, and both weasels and snakes primarily feed on mice. So the weasel and snake have a competitive relationship, just like the $u_{1}$ and $u_{2}$ species in model (1.1), and the eagle is like the $u_{3}$ species in model (1.1), which preys on both weasels ($u_{1}$) and snakes ($u_{2}$). The degradation models of models (1.1)–(1.2) have been intensively investigated [1–10] since 1920s when Lotka [11] and Volterra [12] proposed the classical Lotka-Volterra models. The “functional response” is thought as the core question in these models, which describes the rate at which predators consume prey. In 1989, Arditi and Ginzburg [13] incorporated predator dependence into functional responses, where they regarded the response function as a function of ratio. For example, Holling I, II, and III functional responses depend solely on the volume of prey. However, when predators must search for prey, the response function also depends on the number of predators. Therefore, predator-prey models with ratio-dependent functional response can more accurately reflect objective natural laws in ecosystems. Then, in 1999, Conser et al. [14] showed that it’s more appropriate to consider ratio-dependent terms into predator-prey model by using some basic but different principles. In 2000, authors in [15] constructed a kind of average Lyapunov function to study autonomous RDPPM (1.1) without diffusion, combined with the knowledge of saturated equilibria, the problems of permanent coexistence and extinction are studied of species. In 2009, the sufficient condition for the stability of an autonomous RDPPM with interacting populations was obtained by M. Haque [16]. In 2013, a RDPPM with a strong Allee effect in prey was studied by Gao and Li [17] and it was proved that the system has a Bogdanov-Takens bifurcation related with a catastrophic crash of the predator population. In 2015, Agrawal and Saleem [18] considered a 3-species RDPPM and proved that for the suitable parameters, the system has chaotic attractors. In 2018, Mandal [19] researched a stochastically forced RDPPM with strong Allee effect in prey population and demonstrated that the model has the stable interior equilibrium point or limit cycle for the coexistence of both species. In 2020, by using the comparison principle Jiang et al. [20] investigated the qualitative behaviors of a RDPPM. By utilizing the comparison principle, the global asymptotical stability are studied for the boundary equilibrium, and some sufficient conditions without delays and diffusion effect were obtained. More recently, in 2023, a novel RDPPM with additional food supply was investigated by Yu et al. [21] and the rich dynamic properties of the system was obtained. It is worth noting that the above RDPPM do not include diffusion terms. Due to animals always involuntarily gathering towards food and water sources, a new model obtained by adding diffusion terms to the above systems can more accurately describe the objective laws of population interactions. However, the methods mentioned in the above literature cannot be directly used to study such new models.

In recent years, RDPPM with diffusion has attracted the attention of more and more scientific and technological workers. In 2013, Ko and Ahn [22, 23] studied a diffusion RDPPM incorporating two competing predator and one prey species and achieved the persistence and global attractor of the system. In 2015, Yang et al. [24] studied a diffusion RDPPM with with L-G functional response and achieved some conditions ensuring the existence of coexisting states and attractors for the model with help of the fixed point index theory. In 2017, Wang [25] investigated the dynamical behavior of a homogeneous predator-prey diffusive model with Neumann boundary conditions and Holling type-III functional response and obtained some conditions guaranteeing the existence of non-constant equilibrium solutions and periodic orbits by utilizing coincidence degree theory and bifurcation approach. In 2020, Wu and Zhao [26] studied a predator-prey diffusive model subject to the Allee effect and threshold hunting and analyzed the asymptotically stability of equilibrium point for the model by constructing Jacobian matrix. In 2022, Yan and Zhang [27] studied a predator-prey diffusion model with B-D functional response and achieved the stability and instability criteria of the constant positive equilibrium points for the model. In 2023, Chen and Wu [28] researched the spatiotemporal behavior for a predator-prey diffusion system subject to B-D functional response with help of Leray-Schauder degree theory and Poincare inequalities. It is worth noting that the above models are all autonomous and 2-species models. Due to the fact that the birth rate, death rate, and interaction between population are not invariable, the parameters in an ecosystem should be a function of time rather than a constant. Therefore, the nonautonomous reaction-diffusion RDPPM can better simulate the interaction among species in the predator-prey system. However, it is difficult to research the reaction-diffusion ecosystem of more than 3-species using the eigenvalue methods mentioned in the above literature, and it is even more difficult to study nonautonomous reaction-diffusion ecosystems with functional response.

It is obvious that the stability analysis of a nonautonomous predator-prey diffusion system subject to multi-species is very difficult because the interaction among multi-species is more complex. Based on this, the researches on the field are still open. More and more experts and scholars focus on attention to reaction-diffusion models especially with 3-species recently, but their researches primarily concerned with the competition and mutualism systems without or with delay (cf. [29–35]) as well as the prey-predator systems without ratio-dependent functional responses (cf [36–39]). As is well known, the methods for studying competition and mutualism models are difficult to directly apply to studying predator-prey models, especially nonautonomous multi-species reaction-diffusion predator-prey models. Meanwhile, the introduction of ratio dependent functions greatly increases the research difficulty of such models. Based on the above analysis and inspired by the above work, we will study the global stability of nonautonomous reaction-diffusion RDPPM (1.1)–(1.2) in this paper. It should be noted that Wang et al. [40] researched dynamical behavior of the system (1.1) with feedback controls and without diffusion.

The article organization are showed as follows. In Sect. 2, we will give some definitions and preliminary results. In Sect. 3, we will investigate the existence of the strictly positive SHPS of the nonautonomous reaction-diffusion RDPPM. In Sect. 5, we pay more attention to the globally asymptotically stability of the strictly positive SHPS. In Sect. 6, we will give two numerical examples to support the theoretical findings of this article. Lastly, we will give a conclusion to summarize the important contributions of this article.

Remark 1

The innovations and achievements of this article are listed as follows: (1) By introducing ratio-dependent functional responses, diffusion terms and variable coefficient into the known population models, a new Lotka-Volterra predator-prey system (nonautonomous reaction-diffusion RDPPM) that can more truly depict the interaction among populations is proposed. (2) By considering of comparison principle and fixed point method, some new theories and methods have been creatively developed, the existence of the strictly positive SHPS of the new predator-prey system are obtained in which only a set of simplify verified conditions are needed. (3) By constructing a novel Lyapunov functions and utilizing the approach of UALS for the parabolic partial differential equations, the globally asymptotically stability of the strictly positive SHPS are studied in which some sufficient conditions are obtained. (4) Compared with the results in [22–27, 40], the results obtained in this article are more general, and provides more convenience for the further long-term application of Lotka-Volterra predator-prey system.

2 Preliminary

Some preliminary results and the definition of UALS are showed in this section, the definitions of SHPS and its globally asymptotically stability can be found in reference [41].

Definition 2.1

Suppose that $\tilde{U}(x,t) \equiv (\tilde{u}_{1}(x,t),\tilde{u}_{2}(x,t),\tilde{u}_{3}(x,t))$, $\hat{U}(x,t) = (\hat{u}_{1}(x,t),\hat{u}_{2}(x,t), \hat{u}_{3}(x,t))$, if $\tilde{U}(x,t) \ge \hat{U}(x,t)$ and for $(x,t) \in \Omega \times R^{ +}$

$$\begin{aligned}& \partial \tilde{u}_{1}(x,t)/\partial t - d_{1}(t)\Delta \tilde{u}_{1}(x,t) \\& \quad \ge \tilde{u}_{1}(x,t)[r_{1}(t) - a_{11}(t)\tilde{u}_{1}(x,t) - a_{12}(t)\hat{u}_{2}(x,t) - \frac{a_{13}(t)\hat{u}_{3}(x,t)}{b_{13}(t)\hat{u}_{3}(x,t) + \tilde{u}_{1}(x,t)}], \\& \partial \tilde{u}_{2}(x,t)/\partial t - d_{2}(t)\Delta \tilde{u}_{2}(x,t)\\& \quad \ge \tilde{u}_{2}(x,t)[r_{2}(t) - a_{22}(t)\tilde{u}_{2}(x,t) - a_{21}(t)\hat{u}_{1}(x,t) - \frac{a_{23}(t)\hat{u}_{3}(x,t)}{b_{23}(t)\hat{u}_{3}(x,t) + \tilde{u}_{2}(x,t)}], \\& \partial \tilde{u}_{3}(x,t)/\partial t - d_{3}(t)\Delta \tilde{u}_{3}(x,t) \\& \quad \ge \tilde{u}_{3}(x,t)[ - r_{3}(t) + \frac{a_{31}(t)\tilde{u}_{1}(x,t)}{b_{13}(t)\tilde{u}_{3}(x,t) + \tilde{u}_{1}(x,t)} + \frac{a_{32}(t)\tilde{u}_{2}(x,t)}{b_{23}(t)\tilde{u}_{3}(x,t) + \tilde{u}_{2}(x,t)}], \\& \partial \hat{u}_{1}(x,t)/\partial t - d_{1}(t)\Delta \hat{u}_{1}(x,t)\\& \quad \le \hat{u}_{1}(x,t)[r_{1}(t) - a_{11}(t)\hat{u}_{1}(x,t) - a_{12}(t)\tilde{u}_{2}(x,t) - \frac{a_{13}(t)\tilde{u}_{3}(x,t)}{b_{13}(t)\tilde{u}_{3}(x,t) + \hat{u}_{1}(x,t)}], \\& \partial \hat{u}_{2}(x,t)/\partial t - d_{2}(t)\Delta \hat{u}_{2}(x,t) \\& \quad \le \hat{u}_{2}(x,t)[r_{2}(t) - a_{22}(t)\hat{u}_{2}(x,t) - a_{21}(t)\tilde{u}_{1}(x,t) - \frac{a_{23}(t)\tilde{u}_{3}(x,t)}{b_{23}(t)\tilde{u}_{3}(x,t) + \hat{u}_{2}(x,t)}], \\& \partial \hat{u}_{3}(x,t)/\partial t - d_{3}(t)\Delta \hat{u}_{3}(x,t)\\& \quad \le \hat{u}_{3}(x,t)[ - r_{3}(t) + \frac{a_{31}(t)\hat{u}_{1}(x,t)}{b_{13}(t)\hat{u}_{3}(x,t) + \hat{u}_{1}(x,t)} + \frac{a_{32}(t)\hat{u}_{2}(x,t)}{b_{23}(t)\hat{u}_{3}(x,t) + \hat{u}_{2}(x,t)}], \end{aligned}$$

and

$$\begin{aligned} &\partial \tilde{u}_{i}(x,t)/\partial n \ge 0,\partial \hat{u}_{i}(x,t)/\partial n \le 0,(x,t) \in \partial \Omega \times R^{ +},\tilde{u}_{i}(x,0) \ge u_{i0}(x),\hat{u}_{i}(x,0) \le u_{i0}(x),\\ &\quad x \in \bar{\Omega},i = 1,2,3, \end{aligned}$$

we called $\tilde{U}(x,t)$, $\hat{U}(x,t)$ are a pair of ordered UALS for models (1.1)–(1.2).

Lemma 2.1

([42])

Suppose that $\tilde{U}(x,t)$, $\hat{U}(x,t)$ are a pair of ordered UALS for models (1.1)–(1.2), then there exists a unique solution $U(x,t)$ for models (1.1)–(1.2) such that $\tilde{U}(x,t) \ge U(x,t) \ge \hat{U}(x,t)$.

Lemma 2.2

([43])

If the function $f(t): R^{ +} \to R$ is uniformly continuous, and the limit $\lim _{t \to \infty} \int _{0}^{t} f(s)ds$ exists and is finite, then $\lim _{t \to + \infty} f(t) = 0$.

Lemma 2.3

([44])

Suppose that $V \subset R_{n}$ is compact and convex and the mapping $\varphi :V \to V$ is continuous, then there exists $x^{*} \in V$ such that $\varphi (x^{*}) = x^{*}$.

3 Existence of the SHPS

Suppose that $\varphi (x)$ is ω-periodic function in $R^{ +}$, we denote

$$ \varphi ^{m} = \sup \{ \varphi (x),x \in R^{ +} \},\varphi ^{l} = \inf \left \{ \varphi (x),x \in R^{ +} \right \}. $$

Next, we study the following ODE corresponding to model (1.1)

$$ \left \{ \textstyle\begin{array}{l} \frac{du_{1}(t)}{dt} = u_{1}(t)[r_{1}(t) - a_{11}(t)u_{1}(t) - a_{12}(t)u_{2}(t) - \frac{a_{13}(t)u_{3}(t)}{b_{13}(t)u_{3}(t) + u_{1}(t)}], \\ \frac{du_{2}(t)}{dt} = u_{2}(t)[r_{2}(t) - a_{22}(t)u_{2}(t) - a_{21}(t)u_{1}(t) - \frac{a_{23}(t)u_{3}(t)}{b_{23}(t)u_{3}(t) + u_{2}(t)}], \\ \frac{du_{3}(t)}{dt} = u_{3}(t)[ - r_{3}(t) + \frac{a_{31}(t)u_{1}(t)}{b_{13}(t)u_{3}(t) + u_{1}(t)} + \frac{a_{32}(t)u_{2}(t)}{b_{23}(t)u_{3}(t) + u_{2}(t)}]. \end{array}\displaystyle \right . $$

(3.1)

For the ODE (3.1), we let

$$\begin{aligned}& M_{1}^{*} = \frac{r_{1}^{m}}{a_{11}^{l}},\qquad M_{2}^{*} = \frac{r_{2}^{m}}{a_{22}^{l}},\qquad M_{3}^{1*} = \frac{M_{1}a_{31}^{m} + M_{1}a_{32}^{m} - M_{1}r_{3}^{l}}{r_{3}^{l}b_{13}^{l} - a_{32}^{m}b_{13}^{l}},\\& m_{1}^{*} = \frac{r_{1}^{l}b_{13}^{l} - a_{12}^{m}M_{2}b_{13}^{l} - a_{13}^{m}}{a_{11}^{m}b_{13}^{l}},\\& m_{2}^{*} = \frac{r_{2}^{l}b_{23}^{l} - a_{21}^{m}M_{1}b_{23}^{l} - a_{23}^{m}}{a_{22}^{m}b_{23}^{l}},\qquad m_{3}^{1*} = \frac{a_{31}^{l}m_{1} - (r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}})m_{1}}{(r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}})b_{13}^{m}},\\& M_{3}^{2*} = \frac{M_{2}a_{31}^{m} + M_{2}a_{32}^{m} - M_{2}r_{3}^{l}}{r_{3}^{l}b_{23}^{l} - a_{31}^{m}b_{23}^{l}},\qquad m_{3}^{2*} = \frac{a_{32}^{l}m_{2} - (r_{3}^{m} - \frac{a_{31}^{l}m_{1}}{b_{13}^{m}M_{3}^{2} + m_{1}})m_{2}}{(r_{3}^{m} - \frac{a_{31}^{l}m_{1}}{b_{13}^{m}M_{3}^{2} + m_{1}})b_{23}^{m}}. \end{aligned}$$

Definition 3.1

Suppose that there exist seven positive real numbers $Q_{i},q_{i}, (i = 1,2,3)$ and T, such that $Q_{i} \ge u_{i}(t) \ge q_{i}$, as $t > T$ for each positive solution($u_{1}(t)$, $u_{2}(t)$, $u_{3}(t)$)of the ODE (3.1) with the positive initials, then ODE (3.1) is called permanent.

Theorem 3.1

If it holds that

$$\begin{aligned}& (H_{1}) a_{32}^{m} < r_{3}^{l} < a_{31}^{m} + a_{32}^{m},\qquad (H_{2}) r_{1}^{l}b_{13}^{l} - a_{12}^{m}M_{2}b_{13}^{l} - a_{13}^{m} > 0,\\& (H_{3}) r_{2}^{l}b_{23}^{l} - a_{21}^{m}M_{1}b_{23}^{l} - a_{23}^{m} > 0,\qquad (H_{4}) a_{31}^{l} > r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}} > 0. \end{aligned}$$

Then the ODE (3.1) is permanent.

Proof

When the system (3.1) satisfies the conditions $(H_{1}) - (H_{4})$, we can choose some appropriate positive real numbers $M_{i}, m_{i}, (i = 1,2), M_{3}^{1}, m_{3}^{1}$ such that

$$ 0 < m_{i} < m_{i}^{*} < M_{i}^{*} < M_{i}, i = 1, 2, 0 < m_{3}^{1} < m_{3}^{1*} < M_{3}^{1*} < M_{3}^{1}. $$

(3.2)

According to the first equation of ODE (3.1), it follows that

$$\begin{aligned}& \frac{du_{1}(t)}{dt} \le u_{1}(t)[r_{1}(t) - a_{11}(t)u_{1}(t)] \le u_{1}(t)[r_{{1}}^{m} - a_{{11}}^{l}u_{1}(t)] = a_{{11}}^{l}u_{1}(t)[ - u_{1}(t) + \frac{r_{{1}}^{m}}{a_{{11}}^{l}}] \\& \quad \quad = a_{{11}}^{l}u_{1}(t)[ - u_{1}(t) + M_{1}^{*}] < a_{{11}}^{l}u_{1}(t)[ - u_{1}(t) + M_{1}], \end{aligned}$$

Based on the comparison theorem of ODE, we can obtain

(1) When $0 < u_{1}(t_{0}) < M_{1}$, if $t \ge t_{0}$, then $u_{1}(t) \le M_{1}$.

(2) When $u_{1}(t_{0}) \ge M_{1}$, for a enough large t, one has $u_{1}(t) \le M_{1}$. Otherwise, if $u_{1}(t) > M_{1}$, then there exists $\alpha > 0$ such that $u_{1}(t) \ge M_{1}^{*} + \alpha$. Furthermore, one has

$$ \frac{du_{1}(t)}{dt}\left |_{u_{1}(t) > M_{1}} \right . \le u_{1}(t)[r_{1}(t) - a_{11}(t)u_{1}(t)] \le a_{{11}}^{l}u_{1}(t)[M_{1}^{*} - u_{1}(t)] < - a_{{11}}^{l}\alpha u_{1}(t), $$

thus, it holds that

$$ u_{1}(t) < u_{1}(t_{0})\exp ( - a_{11}^{l}\alpha t) \to 0\text{as} t \to + \infty . $$

The above inequality contradicts $u_{1}(t) > M_{1}$, so we can choose a adequacy large $T_{1} \ge t_{0} \ge 0$ such that

$$ u_{1}(t) \le M_{1}\quad \text{as} t > T_{1}. $$

(3.3)

Similarly, from the second equation of the ODE (3.1), it holds that there exist a enough large $T_{2} \ge t_{0} \ge 0$ such that

$$ u_{2}(t) \le M_{2}\quad \text{as} t > T_{2}. $$

(3.4)

From the third equation of the ODE (3.1), and using (3.3), it follows that

$$\begin{aligned}& \frac{du_{3}(t)}{dt} \le u_{3}(t)[ - r_{3}^{l} + \frac{a_{31}^{m}M_{1}}{b_{13}^{l}u_{3}(t) + M_{1}} + a_{32}^{m}] \\& \quad \quad = u_{3}(t)\frac{ - r_{3}^{l}b_{13}^{l}u_{3}(t) - r_{3}^{l}M_{1} + a_{31}^{m}M_{1} + a_{32}^{m}b_{13}^{l}u_{3}(t) + a_{32}^{m}M_{1}}{b_{13}^{l}u_{3}(t) + M_{1}} \\& \quad \quad = u_{3}(t)\frac{r_{3}^{l}b_{13}^{l} - a_{32}^{m}b_{13}^{l}}{b_{13}^{l}u_{3}(t) + M_{1}}[ - u_{3}(t) + \frac{M_{1}a_{31}^{m} + M_{1}a_{32}^{m} - M_{1}r_{3}^{l}}{r_{3}^{l}b_{13}^{l} - a_{32}^{m}b_{13}^{l}}] \\& \quad \quad = u_{3}(t)\frac{r_{3}^{l}b_{13}^{l} - a_{32}^{m}b_{13}^{l}}{b_{13}^{l}u_{3}(t) + M_{1}}[ - u_{3}(t) + M_{3}^{1*}] < u_{3}(t)\frac{r_{3}^{l}b_{13}^{l} - a_{32}^{m}b_{13}^{l}}{b_{13}^{l}u_{3}(t) + M_{1}}[ - u_{3}(t) + M_{3}^{1}]. \end{aligned}$$

According to the same analysis method as above, we can obtain that

(3) When $0 < u_{3}(t_{0}) < M_{3}^{1}$, if $t \ge t_{0}$, then $u_{3}(t) \le M_{3}^{1}$,

(4) When $u_{3}(t_{0}) \ge M_{3}^{1}$, for a enough large t, one has $u_{3}(t) \le M_{3}^{1}$.

So it follows that there is a enough large $T_{3} \ge t_{0} \ge 0$ such that

$$ u_{3}(t) \le M_{3}^{1}\quad \text{as } t > T_{3}. $$

(3.5)

Next, we prove that $u_{1}(t),u_{2}(t),u_{3}(t)$ have positive lower bound. According to the first equation of ODE (3.1), we can obtain that

$$\begin{aligned}& \frac{du_{1}(t)}{dt} \ge u_{1}(t)[r_{1}^{l} - a_{11}^{m}u_{1}(t) - a_{12}^{m}M_{2} - \frac{a_{13}^{m}}{b_{13}^{l}}] \\& \quad \quad = u_{1}(t)a_{11}^{m}[ - u_{1}(t) + \frac{r_{1}^{l}b_{13}^{l} - a_{12}^{m}M_{2}b_{13}^{l} - a_{13}^{m}}{a_{11}^{m}b_{13}^{l}}] \\& \quad \quad = u_{1}(t)a_{11}^{m}[ - u_{1}(t) + m_{1}^{*}] > u_{1}(t)a_{11}^{m}[ - u_{1}(t) + m_{1}]. \end{aligned}$$

Based on the comparison theorem of ODE, it holds that

(5) When $m_{1} < u_{1}(t_{0})$, if $t \ge t_{0}$, then $m_{1} \le u_{1}(t)$,

(6) When $0 < u_{1}(t_{0}) \le m_{1}$, for a sufficiently large t, one has $m_{1} \le u_{1}(t)$. Otherwise, if $u_{1}(t) < m_{1}$, then there exists $\beta > 0$ such that $u_{1}(t) \le m_{1}^{*} - \beta$. Furthermore, we have

$$ \frac{du_{1}(t)}{dt}\left |_{x_{1}(t) < m_{1}} \right . \ge a_{11}^{m}u_{1}(t)[ - u_{1}(t) + m_{1}^{*}] > a_{11}^{m}\beta u_{1}(t), $$

thus, it holds that

$$ u_{1}(t) > u_{1}(t_{0})\exp (a_{11}^{m}\beta t) \to + \infty \quad \text{as} t \to + \infty . $$

The above inequality contradicts $u_{1}(t) < m_{1}$, so we can choose a adequacy large $T'_{1} \ge t_{0} \ge 0$ such that

$$ u_{1}(t) \ge m_{1}\quad \text{as} t > T'_{1}. $$

(3.6)

Analogously, based on the second equation of the ODE (3.1), we can prove that there exists a enough large constant $T'_{2} > 0$ such that

$$ u_{2}(t) \ge m_{2} > 0, m_{2} < m_{2}^{*} = \frac{r_{2}^{l}b_{23}^{l} - a_{21}^{m}M_{1}b_{23}^{l} - a_{23}^{m}}{a_{22}^{m}b_{23}^{l}}\quad \text{as} t > T'_{2}. $$

(3.7)

According to the third equation of the ODE (3.1), and invoking (3.5), (3.6) and (3.7), it holds that

$$\begin{aligned}& \frac{du_{3}(t)}{dt} \ge u_{3}(t)[ - r_{3}^{m} + \frac{a_{31}^{l}m_{1}}{b_{13}^{m}u_{3}(t) + m_{1}} + \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}}] \\& \quad = u_{3}(t)[\frac{ - (r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}})b_{13}^{m}u_{3}(t) - (r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}})m_{1} + a_{31}^{l}m_{1}}{b_{13}^{m}u_{3}(t) + m_{1}}] \\& \quad = u_{3}(t)\frac{(r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}})b_{13}^{m}}{b_{13}^{m}u_{3}(t) + m_{1}}[ - u_{3}(t) + \frac{a_{31}^{l}m_{1} - (r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}})m_{1}}{(r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}})b_{13}^{m}}] \\& \quad = u_{3}(t)\frac{(r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}})b_{13}^{m}}{b_{13}^{m}u_{3}(t) + m_{1}}[ - u_{3}(t) + m_{3}^{1*}] > u_{3}(t)\frac{r_{3}^{m}b_{13}^{m}}{b_{13}^{m}u_{3}(t) + m_{1}}[ - u_{3}(t) + m_{3}^{1}]. \end{aligned}$$

By the similar analysis method above and the comparison theorem of ODE, it holds that

(7) When $m_{3}^{1} < u_{3}(t_{0})$, if $t \ge t_{0}$, then $m_{3}^{1} \le u_{3}(t)$,

(8) When $u_{3}(t_{0}) \ge m_{3}^{1}$, for a enough large t, we have $m_{3}^{1} \le u_{3}(t)$.

Therefore, we can choose a adequacy large $T'_{3} \ge t_{0} \ge 0$ such that

$$ u_{3}(t) \ge m_{3}^{1}\quad \text{as} t > T'_{3}. $$

(3.8)

From (3.3)–(3.8), and set $T = \max _{1 \le i \le 3}\left \{ T_{i},T'_{i} \right \}$, then we have $m_{i} \le u_{i}(t) \le M_{i}$, $(i = 1,2)$, $m_{3}^{1} \le u_{3}(t) \le M_{3}^{1}$ as $t > T$ for any positive solution($u_{1}(t)$, $u_{2}(t)$, $u_{3}(t)$)of the ODE (3.1) subject to the positive initials. Thus, we complete the proof of Theorem 3.1.

Based on the symmetry of prey species $u_{1}(t)$ and $u_{2}(t)$ in ODE (3.1), the following similar conclusions can be drawn using the same analysis and proof methods mentioned above. □

Theorem 3.2

If it holds that

$$\begin{aligned}& (H_{2}) r_{1}^{l}b_{13}^{l} - a_{12}^{m}M_{2}b_{13}^{l} - a_{13}^{m} > 0,\qquad (H_{3}) r_{2}^{l}b_{23}^{l} - a_{21}^{m}M_{1}b_{23}^{l} - a_{23}^{m} > 0,\\& (H_{5}) a_{31}^{m} < r_{3}^{l} < a_{31}^{m} + a_{32}^{m},\qquad (H_{6}) a_{32}^{l} > r_{3}^{m} - \frac{a_{31}^{l}m_{1}}{b_{13}^{m}M_{3}^{2} + m_{1}} > 0. \end{aligned}$$

Then the ODE (3.1) is permanent.

Theorem 3.3

Suppose that the model (1.1) satisfies the conditions $(H_{1}) - (H_{4})$, then the models (1.1)–(1.2) have a strictly positive SHPS $U(t) = \left ( u_{1}^{*}(t),u_{2}^{*}(t),u_{3}^{*}(t) \right )$.

Proof

Based on the existence and uniqueness theorem of solutions of ODE, we can define a Poincaré operator $\varphi : R_{ +}^{3} \to R_{ +}^{3}$ in the following form

$$ \varphi (U_{0}) = U(t, \omega , t_{0}, U_{0}), $$

where $U(t,\omega ,t_{0},U_{0}) = (u_{1}(t),u_{2}(t),u_{3}(t))$ be a positive solution of the ODE (3.1) subject to the initial conditions $U_{0} = (u_{1}(t_{0}),u_{2}(t_{0}),u_{3}(t_{0}))$. And define

$$ S = \left \{ (u_{1},u_{2},u_{3}) \in R_{ +}^{3}\left | m_{i} \le u_{i} \le M_{i} \right ., i = 1, 2, m_{3}^{1} \le u_{3} \le M_{3}^{1} \right \}, $$

then it is quite clear that $S \subset R_{ +}^{3}$ is a compact and convex set. According to the Theorem 3.1 and the continuity of solution of ODE (3.1) subject to the initial conditions, it is not difficult to know that the mapping T is a continuous mapping from S to S. Furthermore, from Lemma 2.3 we can obtain that the ODE (3.1) has a positive ω-periodic solution $(u_{1}^{ *} (t),u_{2}^{ *} (t),u_{3}^{ *} (t)), t \in R^{ +}$. It is easy to know that ($u_{1}^{ *} (t)$, $u_{2}^{ *} (t)$, $u_{3}^{ *} (t)$)is a strictly positive SHPS for models (1.1)–(1.2). The proof of Theorem 3.3 is completed. □

From Theorem 3.2, the following similar conclusions can be drawn using the same analysis and proof methods used in Theorem 3.3.

Theorem 3.4

Suppose that the model (1.1) satisfies the conditions $(H_{2})$, $(H_{3})$, $(H_{5})$, $(H_{6})$, then the models (1.1)–(1.2) have a strictly positive SHPS $U(t) = \left ( u_{1}^{*}(t),u_{2}^{*}(t),u_{3}^{*}(t) \right )$.

4 Stability of the SHPS for ω-periodic RDPPM (1.1)–(1.2)

In present section, we obtain the globally asymptotically stability of the SHPS for the diffusive nonautonomous ω-periodic RDPPM (1.1)–(1.2) by invoking the method of UALS for the parabolic partial differential equations and Lyapunov stability theory, some easily verifiable sufficient conditions are given.

Theorem 4.1

Assume that the diffusive nonautonomous ω-periodic RDPPM (1.1) satisfies assumptions $(H_{1}) - (H_{4})$ and the following assumptions

$$\begin{aligned}& (H_{7}) a_{{11}}^{l} - a_{{21}}^{m} - \frac{(a_{{13}}^{m} + a_{{31}}^{m}b_{{13}}^{m})M_{3}^{1}}{(b_{{13}}^{l}m_{3}^{1} + m_{1})^{2}} > 0, (H_{8}) a_{{22}}^{l} - a_{{12}}^{m} - \frac{(a_{{23}}^{m} + a_{{32}}^{m}b_{{23}}^{m})M_{3}^{1}}{(b_{{23}}^{l}m_{3}^{1} + m_{2})^{2}} > 0,\\& (H_{9})\frac{ - a_{{13}}^{m}M_{1} + a_{{31}}^{l}b_{{13}}^{l}m_{1}}{(b_{{13}}^{l}m_{3}^{1} + m_{1})^{2}} + \frac{ - a_{{23}}^{m}M_{2} + a_{{32}}^{l}b_{{23}}^{l}m_{2}}{(b_{{23}}^{l}m_{3}^{1} + m_{2})^{2}} > 0, \end{aligned}$$

then there is a strictly positive SHPS $(u_{1}^{ *} (t),u_{2}^{ *} (t),u_{3}^{ *} (t))$. And the SHPS is globally asymptotically stable, i.e., the solution ($u_{1}(x,t)$, $u_{2}(x,t)$, $u_{3}(x,t)$) of the diffusive nonautonomous RDPPM (1.1)–(1.2) with any positive initial values fulfills

$$ \lim _{t \to \infty} \left ( u_{i}(x,t) - u_{i}^{ *} (t) \right ) = 0,\quad \mathrm{uniformly}\ \textit{for} x \in \bar{\Omega},\qquad i = 1,2,3. $$

(4.1)

Proof

By means of Theorem 3.3, we have obtained the existence results, next we pay more attention to the stability. Let $l_{i} = \min _{x \in \bar{\Omega}} u_{i0}(x)$, $r_{i} = \max _{x \in \bar{\Omega}} u_{i0}(x)$, $i = 1,2,3$, then $0 < l_{i} \le u_{i0}(x) \le r_{i}$. Suppose that($\tilde{u}_{1}(t)$, $\tilde{u}_{2}(t)$, $\tilde{u}_{3}(t)$)and($\hat{u}_{1}(t)$, $\hat{u}_{2}(t)$, $\hat{u}_{3}(t)$)are the solution for ODE (3.1) subject to initial values $(\tilde{u}_{1}(0),\tilde{u}_{2}(0),\tilde{u}_{3}(0)) = (r_{1},r_{2},r_{3})$ and $(\hat{u}_{1}(0),\hat{u}_{2}(0),\hat{u}_{3}(0)) = (l_{1},l_{2},l_{3})$ respectively, then there are a pair of ordered UALS($\tilde{u}_{1}(t)$, $\tilde{u}_{2}(t)$, $\tilde{u}_{3}(t)$)and($\hat{u}_{1}(t)$, $\hat{u}_{2}(t)$, $\hat{u}_{3}(t)$)for the diffusive nonautonomous RDPPM (1.1)–(1.2). Therefore, from Lemma 2.1 the diffusive nonautonomous RDPPM (1.1)–(1.2) have a unique solution $(u_{1}(x,t), u_{2}(x,t),u_{3}(x,t)), (x,t) \in \bar{\Omega} \times R^{ +}$, which satisfies

$$ (\hat{u}_{1}(t),\hat{u}_{2}(t),\hat{u}_{3}(t)) \le (u_{1}(x,t),u_{2}(x,t),u_{3}(x,t)) \le (\tilde{u}_{1}(t),\tilde{u}_{2}(t),\tilde{u}_{3}(t)). $$

If we can prove

$$ \lim _{t \to \infty} \tilde{u}_{i}(t) - u_{i}^{ *} (t) = \lim _{t \to \infty} \hat{u}_{i}(t) - u_{i}^{ *} (t) = 0,\qquad (i = 1,2,3), $$

(4.2)

then (4.1) is established. So, if we want to achieve (4.2), we have to prove the solution ($u_{1}(t)$, $u_{2}(t)$, $u_{3}(t)$)for ODE (3.1) with any positive initial value $(u_{1}(0),u_{2}(0),u_{3}(0)) = (u_{10},u_{20},u_{30})$ satisfies

$$ \lim _{t \to \infty} \left ( u_{i}(t) - u_{i}^{ *} (t) \right ) = 0, i = 1,2,3. $$

(4.3)

By means of Theorem 3.1, there exist positive constants $M_{i},m_{i},i = 1,2, M_{3}^{1},m_{3}^{1}$ and T, such that

$$ m_{i} \le u_{i}(t) \le M_{i}, (i = 1,2), m_{3}^{1} \le u_{3}(t) \le M_{3}^{1}\quad \text{for} \ \mathrm{all} t > T. $$

Set Lyapunov function

$$ V(t) = \sum _{i = 1}^{3} [\left | \ln u_{i}(t) - \ln u^{*}_{i}(t) \right |],t > 0. $$

Suppose that $D^{ +} V(t)$ is the right derivation on function $V(t)$, it follows that

$$\begin{aligned}& D^{ +} V(t) \le \sum _{i = 1}^{3} D^{ +} [\left | \ln u_{i}(t) - \ln u^{*}_{i}(t) \right |] = \sum _{i = 1}^{3} {\mathop{\operatorname{sgn}}} \{ u_{i}(t) - u^{*}_{i}(t)\} (\frac{\dot{u}_{i}(t)}{u_{i}(t)} - \frac{\dot{u}^{*}_{i}(t)}{u^{*}_{i}(t)}) \\& \quad \quad \quad = {\mathop{\operatorname{sgn}}} \{ u_{1}(t) - u_{1}^{*}(t)\} [ - a_{11}(t)(u_{1}(t) - u_{1}^{*}(t)) - a_{12}(t)(u_{2}(t) - u_{2}^{*}(t)) \\& - a_{13}(t)(\frac{u_{3}(t)}{(b_{13}(t)u_{3}(t) + u_{1}(t))} - \frac{u_{3}^{*}(t)}{(b_{13}(t)u_{3}^{*}(t) + u_{1}^{*}(t))})] \\& + {\mathop{\operatorname{sgn}}} \{ u_{2}(t) - u_{2}^{*}(t)\} [ - a_{22}(t)(u_{2}(t) - u_{2}^{*}(t)) - a_{21}(t)(u_{1}(t) - u_{1}^{*}(t)) \\& - a_{23}(t)(\frac{u_{3}(t)}{(b_{23}(t)u_{3}(t) + u_{2}(t))} - \frac{u_{3}^{*}(t)}{(b_{23}(t)u_{3}^{*}(t) + u_{2}^{*}(t))})] \\& + {\mathop{\operatorname{sgn}}} \{ u_{3}(t) - u_{3}^{*}(t)\} [a_{31}(t)(\frac{u_{1}(t)}{b_{13}(t)u_{3}(t) + u_{1}(t)} - \frac{u_{1}^{*}(t)}{b_{13}(t)u_{3}^{*}(t) + u_{1}^{*}(t)}) \\& + a_{32}(t)(\frac{u_{2}(t)}{b_{23}(t)u_{3}(t) + u_{2}(t)} - \frac{u_{2}^{*}(t)}{b_{23}(t)u_{3}^{*}(t) + u_{2}^{*}(t)})] \\& = {\mathop{\operatorname{sgn}}} \{ u_{1}(t) - u_{1}^{*}(t)\} [ - a_{11}(t)(u_{1}(t) - u_{1}^{*}(t)) - a_{12}(t)(u_{2}(t) - u_{2}^{*}(t)) \\& - a_{13}(t)\frac{u_{1}(t)(u_{3}(t) - u_{3}^{*}(t)) - u_{3}(t)(u_{1}(t) - u_{1}^{*}(t))}{(b_{13}(t)u_{3}(t) + u_{1}(t))(b_{13}(t)u_{3}^{*}(t) + u_{1}^{*}(t))}] \\& + {\mathop{\operatorname{sgn}}} \{ u_{2}(t) - u_{2}^{*}(t)\} [ - a_{22}(t)(u_{2}(t) - u_{2}^{*}(t)) - a_{21}(t)(u_{1}(t) - u_{1}^{*}(t)) \\& - a_{23}(t)\frac{u_{2}(t)(u_{3}(t) - u_{3}^{*}(t)) - u_{3}(t)(u_{2}(t) - u_{2}^{*}(t))}{(b_{23}(t)u_{3}(t) + u_{2}(t))(b_{23}(t)u_{3}^{*}(t) + u_{2}^{*}(t))}] \\& \quad + {\mathop{\operatorname{sgn}}} \{ u_{3}(t) - u_{3}^{*}(t)\} [a_{31}(t)b_{13}(t)\frac{u_{3}(t)(u_{1}(t) - u_{1}^{*}(t)) - u_{1}(t)(u_{3}(t) - u_{3}^{*}(t))}{(b_{13}(t)u_{3}(t) + u_{1}(t))(b_{13}(t)u_{3}^{*}(t) + u_{1}^{*}(t))} \\& \quad + a_{32}(t)b_{23}(t)\frac{u_{3}(t)(u_{2}(t) - u_{2}^{*}(t)) - u_{2}(t)(u_{3}(t) - u_{3}^{*}(t))}{(b_{23}(t)u_{3}(t) + u_{2}(t))(b_{23}(t)u_{3}^{*}(t) + u_{2}^{*}(t))}] \\& \le \left | u_{1}(t) - u_{1}^{*}(t) \right |[ - a_{11}(t) + a_{21}(t) + \frac{(a_{13}(t) + a_{31}(t)b_{13}(t))u_{3}(t)}{(b_{13}(t)u_{3}(t) + u_{1}(t))(b_{13}(t)u_{3}^{*}(t) + u_{1}^{*}(t))}] \\& \quad + \left | u_{2}(t) - u_{2}^{*}(t) \right |[ - a_{22}(t) + a_{12}(t) + \frac{(a_{23}(t) + a_{32}(t)b_{23}(t))u_{3}(t)}{(b_{23}(t)u_{3}(t) + u_{2}(t))(b_{23}(t)u_{3}^{*}(t) + u_{2}^{*}(t))}] \\& \quad + \left | u_{3}(t) - u_{3}^{*}(t) \right |[\frac{(a_{13}(t) - a_{31}(t)b_{13}(t))u_{1}(t)}{(b_{13}(t)u_{3}(t) + u_{1}(t))(b_{13}(t)u_{3}^{*}(t) + u_{1}^{*}(t))} \\& \quad + \frac{(a_{23}(t) - a_{32}(t)b_{23}(t))u_{2}(t)}{(b_{23}(t)u_{3}(t) + u_{2}(t))(b_{23}(t)u_{3}^{*}(t) + u_{2}^{*}(t))}] \\& \le - \left | u_{1}(t) - u_{1}^{*}(t) \right |[a_{{11}}^{l} - a_{{21}}^{m} - \frac{(a_{{13}}^{m} + a_{{31}}^{m}b_{{13}}^{m})M_{3}^{1}}{(b_{{13}}^{l}m_{3}^{1} + m_{1})^{2}}] \\& \quad - \left | u_{2}(t) - u_{2}^{*}(t) \right |[a_{{22}}^{l} - a_{{12}}^{m} - \frac{(a_{{23}}^{m} + a_{{32}}^{m}b_{{23}}^{m})M_{3}^{1}}{(b_{{23}}^{l}m_{3}^{1} + m_{2})^{2}}] \\& \quad - \left | u_{3}(t) - u_{3}^{*}(t) \right |[\frac{ - a_{{13}}^{m}M_{1} + a_{{31}}^{l}b_{{13}}^{l}m_{1}}{(b_{{13}}^{l}m_{3}^{1} + m_{1})^{2}} + \frac{ - a_{{23}}^{m}M_{2} + a_{{32}}^{l}b_{{23}}^{l}m_{2}}{(b_{{23}}^{l}m_{3}^{1} + m_{2})^{2}}]. \end{aligned}$$

In view of assumptions $(H_{7}) - (H_{9})$, one has

$$\begin{aligned}& \alpha = \min \{ a_{{11}}^{l} - a_{{21}}^{m} - \frac{(a_{{13}}^{m} + a_{{31}}^{m}b_{{13}}^{m})M_{3}^{1}}{(b_{{13}}^{l}m_{3}^{1} + m_{1})^{2}}, a_{{22}}^{l} - a_{{12}}^{m} - \frac{(a_{{23}}^{m} + a_{{32}}^{m}b_{{23}}^{m})M_{3}^{1}}{(b_{{23}}^{l}m_{3}^{1} + m_{2})^{2}},\\& \quad \frac{ - a_{{13}}^{m}M_{1} + a_{{31}}^{l}b_{{13}}^{l}m_{1}}{(b_{{13}}^{l}m_{3}^{1} + m_{1})^{2}} + \frac{ - a_{{23}}^{m}M_{2} + a_{{32}}^{l}b_{{23}}^{l}m_{2}}{(b_{{23}}^{l}m_{3}^{1} + m_{2})^{2}}\} > 0. \end{aligned}$$

Thus,

$$ D^{ +} V(t) \le - \alpha \sum _{i = 1}^{3} \left | u_{i}(t) - u_{i}^{*}(t) \right |. $$

(4.4)

Integrating (4.4) from T to t ($T \ge t_{0}$), it follows that

$$ V(t) + \alpha \int _{T}^{t} (\sum _{i = 1}^{3} \left | u_{i}(t) - u_{i}^{*}(t) \right |) ds \le V(T) < + \infty . $$

Therefore,

$$ \int _{T}^{t} (\sum _{i = 1}^{3} \left | u_{i}(t) - u_{i}^{*}(t) \right | )ds \le \frac{V(T)}{\alpha} < + \infty . $$

(4.5)

By (4.5), we have $\sum _{i = 1}^{3} (\left | u_{i}(t) - u_{i}^{*}(t) \right |) \in L^{1}(T, + \infty )$. From the uniformity permanence of the ODE (3.1), $\sum _{i = 1}^{3} (\left | u_{i}(t) - u_{i}^{*}(t) \right |)$ is uniformity continuous. With help of Lemma 2.2, it follows that

$$ \lim _{t \to + \infty} \left | u_{i}(t) - \right .\left . u_{i}^{*}(t) \right | = 0, (i = 1,2,3). $$

Thus, the proof of Theorem 4.1 is completed. □

From Theorem 3.2 and Theorem 3.4, the following similar conclusions can be drawn using the same analysis and proof methods used in Theorem 4.1.

Theorem 4.2

Assume that the diffusive nonautonomous ω-periodic RDPPM (1.1) satisfies assumptions $(H_{2})$, $(H_{3})$, $(H_{5})$, $(H_{6})$ and the following assumptions

$$\begin{aligned}& (H_{10}) a_{{11}}^{l} - a_{{21}}^{m} - \frac{(a_{{13}}^{m} + a_{{31}}^{m}b_{{13}}^{m})M_{3}^{2}}{(b_{{13}}^{l}m_{3}^{2} + m_{1})^{2}} > 0,\\& (H_{11}) a_{{22}}^{l} - a_{{12}}^{m} - \frac{(a_{{23}}^{m} + a_{{32}}^{m}b_{{23}}^{m})M_{3}^{2}}{(b_{{23}}^{l}m_{3}^{2} + m_{2})^{2}} > 0,\\& (H_{12})\frac{ - a_{{13}}^{m}M_{1} + a_{{31}}^{l}b_{{13}}^{l}m_{1}}{(b_{{13}}^{l}m_{3}^{2} + m_{1})^{2}} + \frac{ - a_{{23}}^{m}M_{2} + a_{{32}}^{l}b_{{23}}^{l}m_{2}}{(b_{{23}}^{l}m_{3}^{2} + m_{2})^{2}} > 0, \end{aligned}$$

then there is a strictly positive SHPS $(u_{1}^{ *} (t),u_{2}^{ *} (t),u_{3}^{ *} (t))$. And the SHPS is globally asymptotically stable, i.e., the solution ($u_{1}(x,t)$, $u_{2}(x,t)$, $u_{3}(x,t)$) of the diffusive nonautonomous RDPPM (1.1)–(1.2) with any positive initial values fulfills $\lim _{t \to \infty} \left ( u_{i}(x,t) - u_{i}^{ *} (t) \right ) = 0$, uniformly for $x \in \bar{\Omega}$, $i = 1,2,3$.

5 Numerical examples

Two examples are given to validate the results achieved in this article. To prove the correctness of the Theorem 4.1 and Theorem 4.2, we choose the 2-periodic function as the coefficients of diffusive nonautonomous ω-periodic RDPPM (1.1)–(1.2).

Example 5.1

Consider the following 3-species reaction-diffusion 2-periodic RDPPM. Based on the assumptions $(H_{1})$–$(H_{4})$ and $(H_{7})$–$(H_{9})$ of Theorem 4.1, with the help of some calculations we choose some special values of parameters shown in models (5.1)–(5.2). It should be noted that, the selection of above parameters is not unique.

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u_{1}(x,t)}{\partial t} - \frac{\partial ^{2}u_{1}(x,t)}{\partial x^{2}} = u_{1}(x,t)[(21 + \cos \pi t) - (13 + \sin \pi t)u_{1}(x,t) \\- (0.075 + 0.025\sin \pi t)u_{2}(x,t) - \frac{(0.065 + 0.035\sin \pi t)u_{3}(x,t)}{(0.95 + 0.05\sin \pi t)u_{3}(x,t) + u_{1}(x,t)}], \\ \frac{\partial u_{2}(x,t)}{\partial t} - \frac{\partial ^{2}u_{2}(x,t)}{\partial x^{2}} = u_{2}(x,t)[(9 + \cos \pi t) - (6.2 + 0.2\sin \pi t)u_{2}(x,t) \\- (0.9 + 0.1\sin \pi t)u_{1}(x,t) - \frac{(0.12 + 0.1\sin \pi t)u_{3}(x,t)}{(0.97 + 0.07\sin \pi t)u_{3}(x,t) + u_{2}(x,t)}], \\ \frac{\partial u_{3}(x,t)}{\partial t} - \frac{\partial ^{2}u_{3}(x,t)}{\partial x^{2}} = u_{3}(x,t)[ - (4.1 + 0.1\cos \pi t) + \frac{(4.07 + 0.03\sin \pi t)u_{1}(x,t)}{(0.95 + 0.05\sin \pi t)u_{3}(x,t) + u_{1}(x,t)} \\ + \frac{(2.05 + 0.05\sin \pi t)u_{2}(x,t)}{(0.97 + 0.07\sin \pi t)u_{3}(x,t) + u_{2}(x,t)}], \\ \quad \quad \quad \quad \quad \quad \quad \quad t > 0,\quad x \in (0,\pi ), \end{array}\displaystyle \right . $$

(5.1)

subject to the following Neumman boundary conditions and initial values

$$ \begin{gathered} \frac{\partial u_{1}}{\partial n} = \frac{\partial u_{2}}{\partial n} = \frac{\partial u_{3}}{\partial n} = 0, t > 0,x = 0, 2\pi , u_{1}(x,0) = 1.6, u_{2}(x,0) = 1.0, \\ u_{3}(x,0) = 0.615, x \in (0, 2\pi ). \end{gathered} $$

(5.2)

By calculating, we have

$$\begin{aligned}& M_{1}^{*} \approx 1.8333, M_{1} = 1.8334, m_{1}^{*} \approx 1.2302, m_{1} = 1.23, \\& M_{2}^{*} \approx 1.6666, M_{2} = 1.6667, M_{3}^{1*} \approx 2.3588, M_{3}^{1} = 2.3589, \\& m_{2}^{*} \approx 0.9253, m_{2} = 0.9252, m_{3}^{1*} \approx 0.1162, m_{3}^{1} = 0.1161, \\& \quad \quad \quad a_{32}^{m} = 2.1 < r_{3}^{l} = 4 < a_{31}^{m} + a_{32}^{m} = 6.2, \\& r_{1}^{l}b_{13}^{l} - a_{12}^{m}M_{2}b_{13}^{l} - a_{13}^{m} \approx 17.75, r_{2}^{l}b_{23}^{l} - a_{21}^{m}M_{1}b_{23}^{l} - a_{23}^{m} \approx 4.55, \\& \quad \quad \quad a_{31}^{l} = 4.1 > r_{3}^{m} - \frac{a_{32}^{l}m_{2}}{b_{23}^{m}M_{3}^{1} + m_{2}} \approx 3.6546 \\& \quad \quad a_{{11}}^{l} - a_{{21}}^{m} - \frac{(a_{{13}}^{m} + a_{{31}}^{m}b_{{13}}^{m})M_{3}}{(b_{{13}}^{l}m_{3}^{1} + m_{1})^{2}} \approx 5.4043, \\& \quad \quad a_{{22}}^{l} - a_{{12}}^{m} - \frac{(a_{{23}}^{m} + a_{{32}}^{m}b_{{23}}^{m})M_{3}^{1}}{(b_{{23}}^{l}m_{3}^{1} + m_{2})^{2}} \approx 0.5204, \\& \frac{ - a_{{13}}^{m}M_{1} + a_{{31}}^{l}b_{{13}}^{l}m_{1}}{(b_{{13}}^{l}m_{3}^{1} + m_{1})^{2}} + \frac{ - a_{{23}}^{m}M_{2} + a_{{32}}^{l}b_{{23}}^{l}m_{2}}{(b_{{23}}^{l}m_{3}^{1} + m_{2})^{2}} \approx 3.6332. \end{aligned}$$

It is quite clear that systems (5.1)–(5.2) satisfy the assumptions of Theorem 4.1. From Theorem 4.1 it is easy to know that the system (5.1) has a strictly positive 2-periodic SHPS $(u_{1}^{ *} (t),u_{2}^{ *} (t),u_{3}^{ *} (t))$. Moreover, the solution ($u_{1}(x,t)$, $u_{2}(x,t)$, $u_{3}(x,t)$) of systems (5.1)–(5.2) fulfills

$\lim _{t \to \infty} \left ( u_{i}(x,t) - u_{i}^{ *} (t) \right ) = 0$, uniformly for $x \in (0,2\pi )$, $i = 1,2,3$.

By employing the finite differences method and the MATLAB 7.1 software package, we can obtain some numerical solutions for the systems (5.1)–(5.2) which are shown in Fig. 1 to Fig. 4. From Figs. 1-4, it is not difficult to find that the systems (5.1)–(5.2) have a strictly positive globally asymptotically stable 2-periodic SHPS.

Figure 1

Evolution process of the density for the species $u_{1}(x,t)$ of systems (5.1)–(5.2)

Figure 2

Evolution process of the density for the species $u_{2}(x,t)$ of systems (5.1)–(5.2)

Figure 3

Evolution process of the density for the species $u_{3}(x,t)$ of systems (5.1)–(5.2)

Figure 4

UOT plane projection of the numerical solutions of systems (5.1)–(5.2)

Example 5.2

Consider the following 3-species reaction-diffusion 2-periodic RDPPM. Based on the assumptions $(H_{2})$, $(H_{3})$, $(H_{5})$, $(H_{6})$ and $(H_{10}) - (H_{12})$ of Theorem 4.2, with the help of some calculations we choose some special values of parameters shown in models (5.3)–(5.4). It should be noted that, the selection of above parameters is not unique.

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u_{1}(x,t)}{\partial t} - \frac{\partial ^{2}u_{1}(x,t)}{\partial x^{2}} = u_{1}(x,t)[(9 + \cos \pi t) - (6.2 + 0.2\sin \pi t)u_{1}(x,t) \\ - (0.075 + 0.025\sin \pi t)u_{2}(x,t) - \frac{(0.065 + 0.035\sin \pi t)u_{3}(x,t)}{(0.97 + 0.07\sin \pi t)u_{3}(x,t) + u_{1}(x,t)}], \\ \frac{\partial u_{2}(x,t)}{\partial t} - \frac{\partial ^{2}u_{2}(x,t)}{\partial x^{2}} = u_{2}(x,t)[(21 + \cos \pi t) - (13 + \sin \pi t)u_{2}(x,t) \\ - (0.9 + 0.1\sin \pi t)u_{1}(x,t) - \frac{(0.12 + 0.1\sin \pi t)u_{3}(x,t)}{(0.95 + 0.05\sin \pi t)u_{3}(x,t) + u_{2}(x,t)}], \\ \frac{\partial u_{3}(x,t)}{\partial t} - \frac{\partial ^{2}u_{3}(x,t)}{\partial x^{2}} = u_{3}(x,t)[ - (4.1 + 0.1\cos \pi t) + \frac{(2.05 + 0.05\sin \pi t)u_{1}(x,t)}{(0.97 + 0.07\sin \pi t)u_{3}(x,t) + u_{1}(x,t)} \\ + \frac{(4.07 + 0.03\sin \pi t)u_{2}(x,t)}{(0.95 + 0.05\sin \pi t)u_{3}(x,t) + u_{2}(x,t)}], \\ \quad \quad \quad \quad \quad \quad \quad \quad t > 0,\quad x \in (0,\pi ) \end{array}\displaystyle \right . $$

(5.3)

with the following Neumman boundary conditions and initial values

$$ \begin{gathered} \frac{\partial u_{1}}{\partial n} = \frac{\partial u_{2}}{\partial n} = \frac{\partial u_{3}}{\partial n} = 0, t > 0,x = 0, 2\pi ,\\ u_{1}(x,0) = 1.4, u_{2}(x,0) = 1.6, u_{3}(x,0) = 0.78, x \in (0, 2\pi ). \end{gathered} $$

(5.4)

By calculating, we have

$$\begin{aligned}& M_{1}^{*} \approx 1.6666, M_{1} = 1.6667, m_{1}^{*} \approx 1.2008, m_{1} = 1.2007, \\& M_{2}^{*} \approx 1.8333, M_{2} = 1.8334, M_{3}^{2*} \approx 2.3588, M_{3}^{2} = 2.3588, \\& m_{2}^{*} \approx 1.2920, m_{2} = 1.2919, m_{3}^{2*} \approx 0.1813, m_{3}^{2} = 0.1812, \\& \quad \quad \quad a_{31}^{m} = 2.1 < r_{3}^{l} = 4 < a_{31}^{m} + a_{32}^{m} = 6.2, \\& r_{1}^{l}b_{13}^{l} - a_{12}^{m}M_{2}b_{13}^{l} - a_{13}^{m} \approx 6.935,r_{2}^{l}b_{23}^{l} - a_{21}^{m}M_{1}b_{23}^{l} - a_{23}^{m} \approx 16.28, \\& \quad \quad \quad \quad a_{32}^{l} = 4.1 > r_{3}^{m} - \frac{a_{31}^{l}m_{1}}{b_{13}^{m}M_{3}^{2} + m_{1}} \approx 3.5428 \\& a_{{11}}^{l} - a_{{21}}^{m} - \frac{(a_{{13}}^{m} + a_{{31}}^{m}b_{{13}}^{m})M_{3}^{2}}{(b_{{13}}^{l}m_{3}^{2} + m_{1})^{2}} \approx 3.7162, a_{{22}}^{l} - a_{{12}}^{m} - \frac{(a_{{23}}^{m} + a_{{32}}^{m}b_{{23}}^{m})M_{3}^{2}}{(b_{{23}}^{l}m_{3}^{2} + m_{2})^{2}} \approx 7.0863, \\& \quad \quad \quad \quad \frac{ - a_{{13}}^{m}M_{1} + a_{{31}}^{l}b_{{13}}^{l}m_{1}}{(b_{{13}}^{l}m_{3}^{2} + m_{1})^{2}} + \frac{ - a_{{23}}^{m}M_{2} + a_{{32}}^{l}b_{{23}}^{l}m_{2}}{(b_{{23}}^{l}m_{3}^{2} + m_{2})^{2}} \approx 3.1008. \end{aligned}$$

It is quite clear that models (5.3)–(5.4) satisfy the assumptions of Theorem 4.1. From Theorem 4.2 it is easy to know that the system (5.3) has a strictly positive 2-periodic SHPS $(u_{1}^{ *} (t),u_{2}^{ *} (t),u_{3}^{ *} (t))$. Moreover, the solution ($u_{1}(x,t)$, $u_{2}(x,t)$, $u_{3}(x,t)$) of models (5.3)–(5.4) fulfills $\lim _{t \to \infty} \left ( u_{i}(x,t) - u_{i}^{ *} (t) \right ) = 0$, uniformly for $x \in (0,2\pi )$, $i = 1,2,3$.

By employing the finite differences method and the MATLAB 7.1 software package, we can obtain some numerical solutions for the systems (5.3)–(5.4) which are shown in Fig. 5 to Fig. 8.

Figure 5

Evolution process of the density for the species $u_{1}(x,t)$ of systems (5.3)–(5.4)

Figure 6

Evolution process of the density for the species $u_{2}(x,t)$ of systems (5.3)–(5.4)

Figure 7

Evolution process of the density for the species $u_{3}(x,t)$ of systems (5.3)–(5.4)

Figure 8

UOT plane projection of the numerical solutions of systems (5.3)–(5.4)

From Figs. 5–8, it is not difficult to find that the systems (5.3)–(5.4) have a strictly positive globally asymptotically stable 2-periodic SHPS.

Studying the conditions under which ecosystems are in equilibrium and how to artificially control them has always been an important topic worthy of in-depth research. From theoretical results (Theorem 4.1 and Theorem 4.2) and numerical simulations (Figs. 1-8), it can be found that the 3-species reaction-diffusion nonautonomous models (1.1)–(1.2) can be in equilibrium when the intrinsic growth rates of two prey species is high enough and the predator’s capture rate is high enough. To be precise, in models (5.1)–(5.2) and (5.3)–(5.4), the densities of prey and predator will oscillate periodically with a period of 2 and distribute homogeneously in space when the time is long enough.

6 Conclusion

This article shows the great strength of UALS approach for nonlinear nonautonomous reaction- diffusion equations. It’s widely used for solving the problems for nonlinear partial differential equations (PDE) in chemistry, engineering and mathematical physics etc. The novel approach constructing Lyapunov function and a pair of ordered UALS provides a reference to deal with stability problem of the nonlinear PDE.

The problem of periodic solution for a 3-species nonautonomous diffusion RDPPM is studied. The existence and stability of the strictly positive SHPS are obtained for the nonautonomous nonlinear reaction-diffusion equations only for some easily verifiable criterions. These criterions improve and generalize some previous results. It is especially worth mention that it’s flexible for applications due to the sufficient conditions obtained in this article are very simple. It should be noted that in this study, we do not considered the delays in the model. However, in ecosystems, time delays are widespread and can affect the stability of the system. Consequently, our next goal is to study the multi-species nonautonomous diffusion RDPPM with time delays.