Comparative analysis of fractional dynamical systems with various operators

1.   Introduction

The model

is used to describe the Nicholson-Bailey host-parasitoid system, where $x_n$ and $y_n$ represent the densities of host and parasitoid at the $n$th generation, respectively, $a$ is the searching efficiency of the parasitoid, $\lambda$ is the host reproductive rate, and $c$ is the average number of viable eggs laid by a parasitoid on a single host. System (1.1) is simple and its positive equilibrium is unstable ^[4,5,19], which indicates that the parasitoid populations, or both the parasitoid and host populations, will go extinct. Therefore this simple model is unrealistic for any practical applications. Up to now, the model has been developed to describe the population dynamic behavior of a coupled host-parasitoid (or predator-prey) system. The improved models display more various dynamic behaviors such as stability, bifurcation, and chaotic phenomenon, see ^{[1,7,8,9,10,16,17,20,21,23]}. For more detailed information, refer to ^{[13,14,15,22,25]}.

As mentioned in ^[18], in many populations it is reasonable to believe that either a refuge exists which isolates some small fraction of the population from density-dependent effects, or that there is a small amount of immigration from outside the system each generation. Therefore, in this paper we consider the system

where

and the initial value $(x_{0}, y_0)\in [0, +\infty)\times [0, +\infty)$. The parameter $\alpha$ is the host reproductive rate at per generation (in the absence of a parasitoid), and the term $\beta$ represents a refuge or a constant amount of immigration of hosts from outside the system per generation.

In ^[12], Kulenović and Ladas proposed an open problem (Open Problem 6.10.16) asking for investigating the global character of all solutions of system (1.2) with parameters $\alpha\in (0, 1)$ and $\beta\in (1, +\infty)$.

Inspired by the aforementioned open problem, in this paper the boundedness, periodic character, transcritical bifurcation, local asymptotic stability, and global asymptotic stability of system (1.2) are discussed under condition (1.3). Our result partially solves the above open problem.

The paper is organized as follows:

Section 1 is the introduction, and Section 2 involves the preliminaries, where some necessary lemmas are presented. Section 3 deals with the boundedness and periodic character of system (1.2). The linearized stability and bifurcation analysis are discussed in Section 4. Section 5 focuses on the global asymptotic stability of the equilibria of system (1.2). Section 6 is the conclusion.

2.   Preliminaries

Prior to commencing the discussion, we present some essential lemmas.

Lemma 2.1. (A Comparison Result ^[11]) Assume that $\alpha\in (0, +\infty)$ and $\beta\in \mathbb{R}$. Let $\{x_n\}_{n = 0}^{\infty}$ and $\{z_n\}_{n = 0}^{\infty}$ be sequences of real numbers such that $x_{0}\le z_{0}$ and

Then $x_n\le z_n$ for $n\ge 0$.

The following lemma is proved in ^[6], which will be applied in analyzing the global attractivity of Eq (2.1). Additionally, one can refer to ^[3,11,12] for further information.

Lemma 2.2. Consider the difference equation

Let $I\subseteq [0, +\infty)$ be some interval and assume that $g\in C[I, (0, +\infty)]$ satisfies the following conditions:

(i) $g(u)$ is non-decreasing in $u$.

(ii) Equation (2.1) has a unique positive equilibrium $\bar{u}\in I$ and the function $g(u)$ satisfies the negative feedback condition:

Then, every positive solution of Eq (2.1) with initial conditions in $I$ converges to $\bar{u}$.

Consider the difference equation

The following strategy for obtaining global attractivity results of Eq (2.2) is derived from ^[12], which is also referenced in ^[2].

Lemma 2.3. Let $[a, b]$ be an interval of real numbers and assume that $G:[a, b]\times [a, b]\rightarrow [a, b]$ is a continuous function satisfying the following properties:

(i) $G(x, y)$ is non-decreasing in $x\in [a, b]$ for each $y\in [a, b]$, and $G(x, y)$ is non-increasing in $y\in [a, b]$ for each $x\in [a, b]$.

(ii) If $(m, M)\in [a, b]\times [a, b]$ is a solution of the system

then $m = M$.

Then, Eq (2.2) has a unique equilibrium $\bar{x}$, and every solution of Eq (2.2) converges to $\bar{x}$.

3.   Boundedness and periodic character

Theorem 3.1. Assume that (1.3) holds. Then every nonnegative solution of system (1.2) is bounded and eventually enters an invariant rectangle $[\beta, \frac{\beta}{1-\alpha}]\times [0, \frac{\alpha\beta}{1-\alpha}]$.

Proof. Using (1.2) and noting that $0 < e^{-y_n}\le 1$ for $y _n\ge 0$, we get

Consider the initial value problem

with initial value $z_0 = x_0$. It follows by Lemma 2.1 that

The solution of Eq (3.1) is given by

and for $n > 1$,

Therefore, the sequence $\{z_n\}$ is decreasing and bounded below by $\frac{\beta}{1-\alpha}$ with the initial value $z_0 > \frac{\beta}{1-\alpha}$, and it is increasing and bounded above by $\frac{\beta}{1-\alpha}$ with the initial value $z_0 < \frac{\beta}{1-\alpha}$, and $z_n = \frac{\beta}{1-\alpha}$ for $n\ge 1$ with the initial value $z_0 = \frac{\beta}{1-\alpha}$. Thus, $\lim\limits_{n\rightarrow \infty}z_n = \frac{\beta}{1-\alpha}$. Hence, for every $\epsilon > 0$, there is an integer $N$ such that, for $n > N$,

and so $x_n\le \frac{\beta}{1-\alpha}$ for $n > N$. Furthermore, when $n > N$,

holds.

Set

Then

Moreover, if $(x_0, y_0)\in [\beta, \frac{\beta}{1-\alpha}]\times[0, \frac{\alpha\beta}{1-\alpha}]$, then

and by using induction, we obtain

So, the rectangle $[\beta, \frac{\beta}{1-\alpha}]\times [0, \frac{\alpha\beta}{1-\alpha}]$ is invariant, which completes the proof. □

Theorem 3.2. Assume that (1.3) holds. Then system (1.2) has no positive prime period-two solution.

Proof. Assume for the sake of contradiction that

is a positive prime period-two solution of system (1.2). Then it should satisfy

and

Clearly, $\xi_1, \xi_2\ge \beta$.

From (3.2) and (3.3), we derive

which are equivalent to

Thus, $\xi_1 = \xi_2\Longleftrightarrow \eta_2 = \eta_1$.

Moreover, (3.2) and (3.3) imply that $\xi_1, \xi_2 > \beta$. This is because, if $\xi_1 = \beta$, then $\xi_2 = 0$ and $\eta_1 = \eta_2 = 0$, which is a contradiction. Similarly, if $\xi_2 = \beta$, then $\xi_1 = 0$ and $\eta_2 = \eta_1 = 0$, which leads to a contradiction as well.

Additionally, combining (3.2), (3.3), and (3.4), we can obtain

and thus

which means that

Set

Then

from which it follows that $A'(t) > 0$ for $t\ge\beta > 0$, and thus $A(t)$ is strictly increasing in $t$ for $t\ge \beta > 0$. So, (3.5) implies that $\xi_1 = \xi_2$. Therefore, $\eta_1 = \eta_2$, a contradiction.

The proof is complete. □

4.   Linearized stability and bifurcation analysis

4.1.   Linearized stability

Theorem 4.1. (i) Assume that (1.3) holds and $\beta\le \frac{1-\alpha}{\alpha}$. Then system (1.2) possesses a unique nonnegative equilibrium $\bar E_x = (\frac{\beta}{1-\alpha}, 0)$.

(ii) Assume that (1.3) holds and $\beta > \frac{1-\alpha}{\alpha}$. Then system (1.2) possesses two equilibria: $\bar E_x = (\frac{\beta}{1-\alpha}, 0)$ and $\bar E = (\bar{x}, \bar{y})\in [\beta, \frac{\beta}{1-\alpha}]\times [0, \frac{\alpha\beta}{1-\alpha}]$.

Proof. The equilibria of system (1.2) can be obtained by solving the following equations:

Clearly, $y = 0$ is always the solution of the second equation of (4.1), and thus $\bar E_x = (\frac{\beta}{1-\alpha}, 0)$ is always the equilibrium of system (1.2).

From the first equation of (4.1), we get

and thus

or, equivalently,

Let

Then, $\phi(0) = 0$, and $\phi(y)\sim y$ as $y\rightarrow +\infty$. Moreover, we have

Let

Then, $\psi'(y) = e^y+\alpha > 0$, from which it follows that the function $\psi(y)$ is strictly increasing in $[0, +\infty)$.

(i) When $\beta\le \frac{1-\alpha}{\alpha}$, $\psi(y) > \psi(0) = 1-\alpha-\alpha\beta\ge 0$ with $y > 0$. Consequently, $\phi'(y) = \frac{1}{e^y}\psi(y) > 0$ for $y > 0$, and system (1.2) has no other equilibrium, which implies that conclusion (i) is valid.

(ii) When $\beta > \frac{1-\alpha}{\alpha}$, $\psi(0) = 1-\alpha-\alpha\beta < 0$, and $\psi(+\infty) = +\infty$. By the continuity of the function $\psi(y)$, there exists a unique root $y^*\in (0, +\infty)$ such that

Hence, $\psi(y) < 0$ with $0 < y < y^*$, and $\psi(y) > 0$ with $y > y^*$. Moreover, $\phi'(y) < 0$ with $0 < y < y^*$, and $\phi'(y) > 0$ with $y > y^*$. It follows that $\phi(y)$ is decreasing in $(0, y^*)$, and $\phi(y)$ is increasing in $(y^*, +\infty)$. Thus, the function $\phi(y)$ attains its minimum at $y^*$, $\phi(y^*) < \phi(0) = 0$, and by the continuity of the function $\phi(y)$, equation $\phi(y) = 0$ has a unique positive root $\bar{y}$ such that $\bar{y} > y^*$.

Adding the two equations of system (4.1) yields

hence

By (4.1) and (4.7), it is easy to obtain that $\bar x\in [\beta, \frac{\beta}{1-\alpha}]$ and $\bar y\in [0, \frac{\alpha\beta}{1-\alpha}]$. Thus, in this case, system (1.2) possesses an additional equilibrium $\bar E = (\bar{x}, \bar{y})$, and conclusion (ii) follows.

The proof is complete. □

Theorem 4.2. (i) Assume that (1.3) holds. Then the equilibrium $\bar E_x = (\frac{\beta}{1-\alpha}, 0)$ is locally asymptotically stable when $\beta < \frac{1-\alpha}{\alpha}$, is nonhyperbolic when $\beta = \frac{1-\alpha}{\alpha}$, and is unstable (a saddle point) when $\beta > \frac{1-\alpha}{\alpha}$.

(ii) Assume that (1.3) holds and $\beta > \frac{1-\alpha}{\alpha}$. Then the unique positive equilibrium $\bar E$ is locally asymptotically stable (a sink).

Proof. Let

By simple calculation, we have

(i) The Jacobian matrix of $F$ evaluated at $\bar E_x$ is given by

and its eigenvalues are

Notice that $\alpha\in(0, 1)$, so $0 < \lambda_1 < 1$, and $0 < \lambda_2 < 1$ with $\beta < \frac{1-\alpha}{\alpha}$, $\lambda_2 = 1$ with $\beta = \frac{1-\alpha}{\alpha}$, and $\lambda_2 > 1$ with $\beta > \frac{1-\alpha}{\alpha}$, which means that result (i) follows.

(ii) The Jacobian matrix of $F$ evaluated at $\bar E$ is given by

and its characteristic equation is

where $p = \alpha e^{-\bar{y}}(1+\bar{x}),$ $q = \alpha^2\bar{x}e^{-\bar{y}}.$

Since the second equation of (4.1) implies that $\bar{x} = \bar{y}/(\alpha(1-e^{-\bar{y}}))$, it can be concluded that

Moreover, noticing that the function $\psi(y)$ defined by (4.4) is strictly increasing in $(0, +\infty)$ and that $\bar{y} > y^*$, we can utilize (4.5) to derive

where $y^*$ is the minimum point of $\phi(y)$. Thus,

In addition, the fact that $\bar y$ is the root of the function $\phi(y)$ given by (4.3) implies that $\phi(\bar{y}) = 0$, namely,

Adding (4.8) and (4.9) yields

From (4.6), we have

and from the first equation of system (4.1), we have

Substituting (4.11) and (4.12) into (4.10) yields

from which it follows that

Applying (4.12), we have $q = \alpha^2\bar{x}e^{-\bar{y}} = \alpha(\bar{x}-\beta)$ and

By the Schur-Cohn criterion, we obtain that $\bar E = (\bar{x}, \bar{y})$ is locally asymptotically stable.

The proof is complete. □

4.2.   Bifurcation analysis

When parameters $\alpha$ and $\beta$ satisfy the condition $\beta = \frac{1-\alpha}{\alpha}$, the equilibrium $\bar E_x = (\frac{\beta}{1-\alpha}, 0)$ is non-hyperbolic with eigenvalue $\lambda_2 = 1$. This indicates a bifurcation probably occurs as the parameter $\beta$ varies and goes through the critical value $\frac{1-\alpha}{\alpha}$. In fact, in this case, a transcritical bifurcation takes place at $\bar E_x$.

Theorem 4.3. Assume that (1.3) holds and let $\beta^* = \frac{1-\alpha}{\alpha}$. Then system (1.2) undergoes a transcritical bifurcation at $\bar E_x$ when the parameter $\beta$ passes through the critical value $\beta^*$.

Proof. Letting $u_{n} = x_n-\frac{\beta}{1-\alpha}$, $v_n = y_n$ shifts the equilibrium $\bar E_x$ to the origin, and tranforms the system (1.2) into

Define $\tau = \beta-\beta^*$ as a small perturbation around $\beta^*$ with $0 < |\tau|\ll1$. Then, the map of system (4.13) can be expressed as:

Expanding (4.14) in a Taylor series at $(u, v, \tau) = (0, 0, 0)$ gives

where

and $O(3)$ is the sum of all remainder terms with a frequency greater than 3.

Let

be an invertible matrix. Through the variable transformation

the map (4.15) is transformed into the form

where

By the center manifold Theorem 2.1.4 in ^[24], for the map (4.16), there exists a center manifold that can be locally represented in the form:

for $\delta$ sufficiently small. Suppose that the center manifold has the representation

Then, it satisfies

where $O(2)$ represents the sum of all remainder terms with a frequency greater than 2. Hence,

Comparing the corresponding coefficients of terms in Eq (4.17), we have

so the map (4.16) on the center manifold can be written as

Since

therefore a transcritical bifurcation takes place at the equilibrium $(Y, \omega) = (0, 0)$ of the map (4.16), implying that, as the parameter $\beta$ changes and passes through the critical value $\beta^*$, system (1.2) undergoes a transcritical bifurcation at $\bar E_x$.

The proof is complete. □

5.   Global asymptotic stability

In view of Lemma 4.2, to deal with the global asymptotic stability of $E_x$ and $\bar E$, it is sufficient to solve its global attractivity.

Consider the difference equation

with $A\in (0, +\infty)$ and the initial value $u_0\in [0, +\infty)$.

Lemma 5.1. When $A\le 1$, Eq (5.1) possesses a unique equilibrium zero, and when $A > 1$, an additional positive equilibrium $\bar{u}$ emerges satisfying $\bar{u} > \ln A$.

Proof. Clearly, zero is always an equilibrium of Eq (5.1). The positive equilibrium can be obtained by solving the equation

Let

Then, $h(0) = 0$, $h(+\infty) = -\infty$, and $h'(u) = A e^{-u}-1$, $h''(u) = -Ae^{-u} < 0$.

When $A\le 1$, $h'(u) < h'(0) = A-1\le 0$ for $u > 0$, and thus Eq (5.1) has a unique equilibrium, namely zero.

When $A > 1$, the function $h(u)$ attains its maximum at $u = \ln A$. Hence, by the continuity of $h(u)$, there exists a unique $\bar{u}$ such that $h(\bar{u}) = 0$, namely Eq (5.1) has a unique positive equilibrium $\bar{u}$ which satisfies $\bar{u} > \ln A$ and $h(u) > 0$ for $0 < u < \bar{u}$, and $h(u) < 0$ for $u > \bar{u}$. □

Lemma 5.2. (i) Assume that $A\le 1$. Then every nonnegative solution of Eq (5.1) converges to the zero equilibrium.

(ii) Assume that $A > 1$. Then every positive solution of Eq (5.1) converges to the unique positive equilibrium $\bar{u}$.

Proof. (i) Clearly, $u_n = 0$ with $u_0 = 0$ for $n\ge 0$, and the result follows. Given $u_0 > 0$, then $u_n > 0$ for $n\ge 1$, and

from which it follows by induction that the sequence $\{u_n\}$ is strictly decreasing and bounded below by zero, so it is convergent. Since, in this case Eq (5.1) has a unique equilibrium zero, hence $\lim\limits_{n\rightarrow \infty}u_n = 0$.

(ii) Let $g(u) = A(1-e^{-u})$. Observing that the function $g(u)$ is increasing for $u > 0$, and using the properties of the function $h(u)$ defined by (5.2), we obtain

and

Hence,

and condition (ii) in Lemma 2.2 is satisfied. It follows that $\lim\limits_{n\rightarrow \infty}u_n = \bar{u}$ with $u_0 > 0$. □

We now start the discussion of our main results.

Theorem 5.3. Every solution $\{(x_n, y_n)\}$ of system (1.2) with $x_0y_0 = 0$ converges to $\bar E_x$.

Proof. Notice that $y_1 = 0$ with $x_0 = 0$, so it is sufficient to discuss the case that $y_0 = 0$. Obviously, in this case, $y_n = 0$ for $n\ge 1$, and thus system (1.2) becomes

and

since $\alpha\in (0, 1)$, finishing the proof. □

Theorem 5.4. Assume that (1.3) holds and $\beta\le \frac{1-\alpha}{\alpha}$. Then the unique equilibrium $\bar E_x$ of system (1.2) is a global attractor of all nonnegative solutions.

Proof. Let $\{(x_n, y_n)\}$ be a nonnegative solution of system (1.2). Then, from Theorem 3.1, the subsequence $\{x_n\}$ is eventually bounded and thus there exists an integer $N$ such that $x_n\le \frac{\beta}{1-\alpha}$ for $n > N$. Using the second equation of system (1.2), we get

Noticing that, in this case $\frac{\alpha\beta}{1-\alpha}\le 1$ and applying 5.2 (i), we obtain that every nonnegative solution of the difference equation

converges to zero. Using the boundedness of the subsequence $\{y_n\}$, (5.3) yields

from which it follows that $\lim\limits_{n\rightarrow \infty} y_n = 0$ and $\lim\limits_{n\rightarrow \infty} x_n = \frac{\beta}{1-\alpha}$. Thus, $\lim\limits_{n\rightarrow \infty} (x_n, y_n) = \bar E_x$.

The proof is complete. □

In view of Theorems 5.4 and 4.2 (i), we have the following result:

Theorem 5.5. Assume that (1.3) holds and $\beta < \frac{1-\alpha}{\alpha}$. Then the unique equilibrium $\bar E_x$ of system (1.2) is globally asymptotically stable.

Next, we deal with the global asymptotic stability of the unique positive equilibrium $\bar E$. We will provide a sufficient condition for $\bar E$ to be globally asymptotically stable with respect to all positive solutions $\{(x_n, y_n)\}$ of system (1.2). The positive solution we talk about here means a solution of system (1.2) satisfying $x_n, y_n > 0$ for $n\ge 0$.

Theorem 5.6. Assume that (1.3) holds and $\beta\ge \frac{1+\alpha}{\alpha}$. Then the unique positive equilibrium $\bar E$ of system (1.2) is a global attractor of all positive solutions.

Proof. Let $\{(x_n, y_n)\}$ be a solution of system (1.2) with $x_0y_0\ne 0$, then $y_n > 0$ for $n\ge 1$.

From the second equation of system (1.2), we get

then

or, equivalently,

which is a second-order difference equation with initial values $y_1 = \alpha x_0(1-e^{-y_0})$, $y_0 > 0$.

Clearly, the equilibrium of Eq (5.4) is not equal to zero and it must satisfy the equation

which is the equation defined by (4.2). Hence, Eq (5.4) has a unique positive equilibrium, namely $\bar{y}$.

Equation (5.4) implies that

If $\beta\ge \frac{1+\alpha}{\alpha}$, then $\alpha\beta > 1$. By utilizing Lemma 5.2 (ii), it can be concluded that every positive solution of the difference equation

converges to its positive equilibrium, denoted by $\tilde{y}$, and by Lemma 5.1, $\tilde{y} > \ln \alpha\beta$. Hence, for $\epsilon = \tilde{y}-\ln \alpha\beta > 0$, there exists an integer $N$ such that $\tilde{y}_n > \tilde{y}-\epsilon = \ln \alpha\beta$ for $n > N$. Further, $y_{n}\ge \tilde{y}_{n} > \ln \alpha\beta > 0$ for $n > N.$ Therefore,

In view of Theorem 3.1, it follows that every positive solution of Eq (5.4) eventually enters an invariant interval $[\ln \alpha\beta, \frac{\alpha\beta}{1-\alpha}]\subset [0, \frac{\alpha\beta}{1-\alpha}]$, and $\bar{y}\in [\ln \alpha\beta, \frac{\alpha\beta}{1-\alpha}]$ is unique.

Set

then $G$ is increasing in $u$ for $v > 0$, and is decreasing in $v$ for $u > 0$.

Let $(m, M)\in [\ln \alpha\beta, \frac{\alpha\beta}{1-\alpha}]\times [\ln \alpha\beta, \frac{\alpha\beta}{1-\alpha}]$ be a solution of the following system:

Then we have

Adding (5.6) and (5.7) yields

which is equivalent to

Consider the function

To prove that $m = M$, it is sufficient to show that the function $I(t)$ is injective on the interval $[\ln \alpha\beta, \frac{\alpha\beta}{1-\alpha}]$ under the condition that $\beta\ge \frac{1+\alpha}{\alpha}$. Simple computation shows that

Let

then

and so, for $t > 0$,

Therefore, $I'(t) > 0$ for $t > 0$, which implies that the function $I(t)$ is strictly increasing on the interval $[\ln \alpha\beta, \frac{\alpha\beta}{1-\alpha}]$. Thus, equality (5.8) yields $m = M$. By applying Lemma 2.3, we get that every positive solution of Eq (5.4) converges to $\bar{y}$.

Consequently, every positive solution of system (1.2) satisfies $\lim\limits_{n\rightarrow \infty}y_n = \bar{y}$ and $\lim\limits_{n\rightarrow \infty}x_n = \bar{x}$, and so $\lim\limits_{n\rightarrow \infty} (x_n, y_n) = \bar {E}$.

The proof is complete. □

In view of Theorems 5.6 and 4.2 (ii), we have the following result:

Theorem 5.7. Assume that (1.3) holds and $\beta\ge\frac{1+\alpha}{\alpha}$. Then the unique positive equilibrium $\bar E$ of system (1.2) is globally asymptotically stable.

6.   Conclusions

In this work, the global behavior of a discrete population model (1.2) is considered with the conditions $\alpha\in (0, 1)$, $\beta\in (0, +\infty)$. It is shown that, for all $\alpha\in (0, 1)$ and $\beta\in (0, +\infty),$ every nonnegative solution of this system is bounded and there is no positive prime period-two solution. However, the existence of equilibria, the local stability, bifurcation, and the global asymptotic stability depend upon the parameters $\alpha, \beta$. Specifically, if $\beta\le \frac{1-\alpha}{\alpha}$, then this system possesses a unique equilibrium $\bar E_x$. It is globally asymptotically stable for $\beta < \frac{1-\alpha}{\alpha}$, and as parameter $\beta$ varies and passes through the critical value $\frac{1-\alpha}{\alpha}$, this system experiences a transcritical bifurcation at $\bar E_x$. If $\beta > \frac{1-\alpha}{\alpha}$, then this system possesses two equilibria, $\bar E_x$ and $\bar E$, where $\bar E_x$ is unstable and $\bar E$ is locally asymptotically stable. Finally, a sufficient condition $\beta\ge \frac{1+\alpha}{\alpha}$ is established, under which $\bar E$ is globally asymptotically stable.

The research result indicates that the use of refuge or external immigration of hosts can contribute to stabilizing the system. If the level of the use of refuge or external immigration of hosts per generation remains at or below the threshold $\frac{1-\alpha}{\alpha}$, the parasitoids will go extinction for all initial populations. Once this threshold is surpassed, the extinct equilibrium $\bar E_x$ loses its stability and the stable coexistence equilibrium $\bar E = (\bar x, \bar y)$ emerges. Specifically, maintaining the level at or above $\frac{1+\alpha}{\alpha}$ guarantees the hosts and the parasitoids will eventually coexist at a steady density $(\bar x, \bar y)$ for all initial populations. Therefore, it is essential to keep enough of a level of refuge or external immigration of hosts for the long-term survival and stability of this system.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 11701425, 12261078). The second author acknowledges the support of the Longyuan Youth Talent Project of Gansu Province.

Conflict of interest

All authors declare that there are no conflicts of interests regarding the publication of this paper.