Positive periodic stability for a neutral-type host-macroparasite equation

1.   Introduction

In general, a neutral-type functional differential equation (NFDE) is one in which the derivatives of the past history or derivatives of functions of the past history are involved, as well as the present state of the system. In ^[1], Hale and Verduyn Lunel introduced the definition and basic theories of NFDE. NFDE has wide applications in many areas, including non-destructive transmission line systems, neural networks, population models; see ^{[2,3,4,5,6,7,8,9,10,11,12]} and related papers.

In 1995, May and Anderson ^[13] first studied the host-macroparasite model as follows:

where $x(t)$ denotes the number of sexually mature worms in the human community. The means of other parameters can be found in ^[13]. Elabbasy et al. ^[14] further considered the oscillation properties of Eq (1.1). Saker and Alzabut ^[15] established the existence, global attractivity, and oscillation of a positive periodic solution to an impulsive delay host-macroparasite model by using the continuation theorem of coincidence degree and the Lyapunov functional method. Yao ^[16] studied the existence and global exponential stability of an almost periodic solution for Eq (1.1) on time scales by using the contraction mapping fixed point theorem, exponential dichotomy, and Gronwall inequality. Yao ^[17] also studied the existence and exponential stability of almost periodic solutions for a difference host-macroparasite equation. Due to being in a changing environment, the coefficients of a population model should be continuously changing functions. To be more accurate, the coefficients and delays in population models can be periodically time-varying. To the best of the authors' knowledge, there are few results of positive periodic solutions for the neutral-type host-macroparasite model.

Motivated by the above work, in the present paper, we study the following nonautonomous neutral-type host-macroparasite equation with multiply time-varying delays:

where $|c|\neq1$ and $\tau, N_i > 0$ are constants, $a(t), b_i(t), d_i(t), \gamma_i(t) > 0$ are $T$-periodic functions.

We list the main innovations of this paper:

(1) We first study positive periodic stability for a neutral-type host-macroparasite equation and generalize the existing results for host-macroparasite model. A neutral-type equation has richer dynamic behaviors.

(2) We develop topological degree theory for studying positive periodic stability. The research method in this article can also be used to study other types of neutral equations.

(3) We innovatively use the properties of neutral-type operators for studying host-macroparasite models.

The remaining framework of this paper is organized as follows: We study the existence of positive periodic solutions of Eq (1.2) in Section 2. Section 3 gives the globally asymptotic and exponential stability of Eq (1.2). Section 4 gives a numerical example for verifying our results. We draw some conclusions in Section 5.

In the whole paper, we use the notations:

where $f$ is a bounded continuous function on $\mathbb{R}$.

2.   Existence of positive periodic solutions

The properties of the neutral-type operator are crucial for the proof of our main results. We give the following lemma:

Lemma 2.1. (^[18,19]) Define $A$ on $C_T$ by

where $C_T = \{x:x\in C(\mathbb{R}, \mathbb{R}), x(t+T)\equiv x(t)\}, \; c$ and $\tau > 0$ are constants. If $|c|\neq 1$, then $A$ has a unique continuous bounded inverse $A^{-1}$ satisfying

Obviously, we have

(1) $||A^{-1}||\le \frac{1}{|1-|c||}$;

(2) $\int_0^T|(A^{-1}f)(t)|dt\le\frac{1}{|1-|c||}\int_0^T|f(t)|dt, \forall f\in C_T$;

(3) $\int_0^T|(A^{-1}f)(t)|^2dt\le\frac{1}{|1-|c||}\int_0^T|f(t)|^2dt, \forall f\in C_T$.

Based on Lemma 2.1, Eq (1.2) can be written by the following equation:

Obviously, Eq (1.2) is equivalent to Eq (2.2). The existence of periodic solutions of Eq (2.2) will be proved as a consequence of a generalized continuation theorem; see Theorem 6.3 in ^[20]. In our case, we consider

Lemma 2.2. Suppose that there exists an open bounded set $\Omega\subset C_T$ such that

(ⅰ) The equation

has no solutions on $\partial\Omega$, where $\lambda\in(0, 1)$;

(ⅱ) For $Ax\in\partial\Omega\cap\mathbb{R}$,

(ⅲ) $deg_B(g, \Omega\cap\mathbb{R}, 0)\neq0.$

Then, Eq (2.2) has at least one periodic solution $Ax$ on $\Omega$.

Remark 2.1. If $Ax$ is a periodic solution of Eq (2.2), let $Ax = y$; by Lemma 2.1, then $x = A^{-1}y$ is a periodic solution of Eq (2.2).

For obtaining the existence of a positive periodic solution of Eq (2.2) by using the topological degree theory, we need to find an priori bound for any periodic solution of Eq (2.2).

We need the following assumption:

(H$_1$) $\frac{1}{a^+} \sum_{i = 1}^n\frac{b_i^-\frac{1}{N_id_i^+}}{[1+\frac{1}{N_i}]^{N_i+1}} -\frac{cR_0}{1-c} > 0$, where $R_0$ is defined by Lemma 2.3.

Lemma 2.3. If $c\in[0, 1)$, then every non-negative $T$-periodic solution $(Ax)(t)$ of Eq (2.3) is bounded above for each $\lambda\in(0, 1)$.

Proof: Let $(Ax)(t)$ be an non-negative $T$-periodic solution of Eq (2.3) and $(Ax)^+ = (Ax)(\xi) = R,$ where $\xi\in[0, T]$. If $(Ax)(t)\geq0$, by Lemma 2.1 and $c\in[0, 1)$, then

It follows by $(Ax)'(\xi) = 0$ that

In view of (2.4), then

Consider the function

Since $f'(x) = \frac{1-d_i^-N_ix}{(1+d_i^-x)^{N_i+2}}$, $f(x)$ is increasing on $[0, \frac{1}{N_id_i^-}]$ and decreasing on $[\frac{1}{N_id_i^-}, +\infty)$. Hence,

From (2.5) and (2.6), we have

Remark 2.2. Using $(Ax)(t) < R_0$ for $t\in \mathbb{R}$ and Lemma 2.1, we have

Lemma 2.4. If assumption (H$_1$) holds and $c\in[0, 1)$, then every non-negative $T$-periodic solution $(Ax)(t)$ of Eq. (2.3) is bounded below for each $\lambda\in(0, 1)$.

Proof: Let $(Ax)(t)$ be a non-negative $T$-periodic solution of Eq (2.3) and $(Ax)^- = (Ax)(\eta) = r,$ where $\eta\in[0, T]$. It follows by $(Ax)'(\eta) = 0$ that

In view of (2.4) and Remark 2.2, then

i.e.,

Take the maximum value on both sides of the above inequality; by assumption (H$_1$), then,

thus,

Remark 2.3. Since $(Ax)(t) > r_0$ for all $t\in\mathbb{R}$, by Lemma 2.1, we have

Now, we show the existence of at least one positive periodic solution of Eq (1.2).

Theorem 2.1. Assume that assumption (H$_1$) holds and $c\in[0, 1)$. Then, Eq (1.2) has at least one positive $T$-periodic solution.

Proof: The proof of this result is based on Lemma 2.2. Since assumption (H$_1$) holds, from Lemmas 2.3 and 2.4, the periodic solutions of Eq (2.3) exists lower and upper bounds for all $\lambda\in(0, 1)$. Define the set $\Xi\subset C_T$ by

where positive constants $r_0$ and $R_0$ are defined by Lemmas 2.3 and 2.4, respectively. From Lemmas 2.3 and 2.4, condition (ⅰ) of Lemma 2.2 holds. Next, we prove that condition (ⅱ) of Lemma 2.2 holds. For $Ax\in\partial\Xi$, if $Ax = r_0$ with $r > r_0$, we have

Let $r\rightarrow r_0$ in the above inequality, we have $g(r_0) > 0$. On the other hand, if $Ax = R_0$ with $R < R_0$, we have

Let $R\rightarrow R_0$ in the above inequality, we have $g(R_0) < 0$. Hence, condition (ⅱ) of Lemma 2.2 holds. It remains to show that condition (ⅲ) of Lemma 2.2 holds. Define the map $\mathcal{H}(x, \mu):\mathbb{R}\times[0, 1]$ by

Obviously, $\mathcal{H}(x, \mu)$ does not vanish on $\partial\Xi$ for any $\mu\in[0, 1]$. So, we have

Applying Lemma 2.2, Eq (2.2) has at least one positive $T$-periodic solution $(Ax)(t)$. Let $(Ax)(t) = y(t)$, then Eq (1.2) has at least one positive $T$-periodic solution $(A^{-1}y)(t)$.

Remark 2.4. Mawhin's continuation and its generalizations are often used to study the existence for periodic solutions for functional differential equations, see ^{[21,22,23,24,25]}. However, when studying the existence of positive periodic solutions using the above theorem, it is very difficult to estimate the prior bound of the solution. When proving Theorem 2.1, we used mathematical analysis methods to estimate the range of positive periodic solutions, thereby obtaining the existence of positive periodic solutions.

Remark 2.5. The proof of Lemma 2.2 can be obtained by Mawhin's continuation theorem.

Mawhin's continuation theorem ^[26]: Suppose that X and Y are two Banach spaces, and $L:D(L)\subset X\rightarrow Y,$ is a Fredholm operator with index zero. Furthermore, $\Omega\subset X$ is an open bounded set and $N:\bar{\Omega}\rightarrow Y$ is $L$-compact on $\bar{\Omega}$. if all the following conditions hold:

(1) $Lx\neq\lambda Nx, \forall x\in\partial\Omega\cap D(L), \forall \lambda\in(0, 1),$

(2) $Nx\notin ImL, \forall x\in\partial\Omega\cap KerL,$

(3) $deg_B\{QN, \Omega\cap KerL, 0\}\neq 0$.

Then, equation $Lx = Nx$ has a solution on $\bar{\Omega}\cap D(L).$

Let

Then, Lemma 2.2 is similar to Mawhin's continuation theorem.

3.   Stability of positive periodic solution

In this section, we will deal with the globally asymptotic and exponential stability of Eq (1.2). As is general in the literature on population models, our asymptotic results are obtained by constructing appropriate Lyapunov functionals. Particularly, we define the region of stability of the solutions of Eq (1.2) as the following set:

To reach our stability results, we make the following assumptions:

(H$_2$) The delays involved in Eq (1.2) are continuously differentiable and satisfy:

(H$_3$) $\frac{a^-(1-2c)}{1-c} -\frac{1}{1-c}\sum_{i = 1}^n\frac{b_i^+}{1-\gamma_i^*} > 0.$

We first state and prove the globally asymptotic theorem.

Theorem 3.1. Assume that assumptions (H$_1$)–(H$_3$) hold and $c\in[0, 1)$. Then, Eq (1.2) has a unique asymptotically stable $T$-periodic solution.

Proof: Let $(Ax)(t), (Ay)(t)\in\Gamma$ be two solutions of Eq (2.2). Consider the following functional:

Calculating the upper right Dini derivative of $V(t)$ along the solutions of Eq (2.2), we have

Consider the function

Since $\tilde{f}'(x) = \frac{1-d_i(t)N_ix}{(1+d_i(t)x)^{N_i+2}}$, then $|\tilde{f}'(x)|\leq1$. From mean value theorem, we have

By assumption (H$_2$), we have

In view of (3.1)–(3.3) and Lemma 2.1, we have

It follows by assumption (H$_3$) and (3.4) that there exists a positive constant $\alpha$ such that

then,

and

Since $|(Ax)(s)-(Ay)(s)|\in L^1([0, \infty))$, by Barbalat's Lemma ^[27], we have

Using $|x(t)-y(t)|\leq\frac{1}{1-c}|(Ax)(t)-(Ay)(t)|$, we also have

Hence, all solutions of the Eq (1.2) in $\Gamma$ converge to a $T$-periodic solution and there exists a unique periodic solution of Eq (1.2) in $\Gamma$.

Now, we show the globally exponential stability of Eq (1.2). Let

By assumption (H$_3$), we have $F(0) > 0$. From the continuity of $F$, there exists a constant $\lambda_0$ such that

Theorem 3.2. Assume that assumptions (H$_1$)–(H$_3$) hold and $c\in[0, 1)$. Then, each $T$-periodic solution of Eq (1.2) is globally exponentially stable.

Proof: Let $(Ax)(t), (Ay)(t)\in\Gamma$ be two solutions of Eq (2.2). Consider the following Lyapunov functional:

Calculating the upper right Dini derivative of $\Phi(t)$ along the solutions of Eq (2.2), we have

From (3.5), (3.7), assumptions (H$_1$)–(H$_3$), and Lemma 2.1, we have

Thus,

Let $\lambda = \lambda_0$, in view of (3.6), we have

Hence, $\Phi(t)$ is decreasing for all $t\geq0$ along the solutions of Eq (2.2); consequently, we get

and

Using Lemma 2.1, we have

Hence, each $T$-periodic solution of Eq (1.2) is globally exponentially stable.

Remark 3.1. The proofs of Theorems 3.1 and 3.2 are based on proper Lyapunov functions and the properties of neutral-type operators. It should be pointed out that the Lyapunov functions of this paper are different from those of the related papers; see ^[7,15,16,17].

4.   An example

For Eq (1.2), let

Regard the scalar neutral-type host-macroparasite equation as follows:

After simper computations, we obtain

Then, we have

Hence, Eq (4.1) satisfies all assumptions in Theorem 2.1, and then it follows that Eq (4.1) has at least one positive periodic solution. In Figure 1, we give the numerical simulations of positive periodic solutions to Eq (4.1) with different initial conditions $x(0) = 1.615, \; x(0) = 1.421$, and $x(0) = 1.171$.

5.   Conclusions

Neutral-type functional differential equations are more complex equations compared to functional differential equations, which contain rich dynamical behaviors. In this paper, we deal with positive periodic solutions of a neutral-type host-macroparasite equation. First, based on a generalized continuation theorem, some sufficient conditions have been proposed to ensure the existence of a positive periodic solution. Then, by the Lyapunov-Krasovskii functional method and some inequality techniques, the asymptotic properties of the positive periodic solution are addressed. Finally, we give a numerical example for verifying the correctness of the results obtained.

Among the projections of this work, we will concentrate on the possible extension of the present study to a neutral-type host-macroparasite equation on time scales, which can unify discrete and continuous equations.

Author contributions

Axiu Shu: Methodology, writing-review and editing; Xiaoliang Li: Supervision, methodology; Bo Du: Writing-original draft; Tao Wang: Formal analysis. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

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

Acknowledgements

The authors would like to express their sincere thanks to the editor and anonymous reviewers for constructive comments and suggestions to improve the quality of this paper.

This work is supported by the Doctor Training Program of Jiyang College, Zhejiang Agriculture and Forestry University (RC2022D03); Anhui Province University Research Project (Key Project) (2024AH051122); Qinglan Project of Jiangsu Province of China (2022); and Huai'an City Science and Technology Project (HAB202357).

Conflict of interest

The authors declare no conflicts of interest.