The uniqueness of limit cycles in a predator-prey system with Ivlev-type group defense

1.   Introduction

The main objective of the paper is to investigate the qualitative behavior of a differential equation

where $\vartheta \geq \vartheta _{0} > 0$ and $\eta$ is a positive integer.

Let us define the corresponding function $\upsilon$ for the solution $x$ as follows:

Furthermore, we make the following supposition:

${\bf{(M1)}}$ $\alpha$ and $\beta \ $are quotients of odd positive integers;

${\bf{(M2)}}$ $b\in C\left(\left[\vartheta _{0}, \infty \right), \left(0, \infty \right) \right),$ $b^{\prime }\left(\vartheta \right) \geq 0\ $and

${\bf{(M3)}}$ $\tau, \delta _{i}\in C\left(\left[\vartheta _{0}, \infty \right), \mathbb{R} \right),$ $\delta _{i}\left(\vartheta \right) \leq \vartheta$, $\tau \left(\vartheta \right) \leq \vartheta$, $\tau ^{\prime }\left(\vartheta \right) > 0,$ and $\lim_{\vartheta \rightarrow \infty }\tau \left(\vartheta \right) = \lim_{\vartheta \rightarrow \infty }\delta _{i}\left(\vartheta \right) = \infty,$ $i = 1, 2, ...\eta;$

${\bf{(M4)}}$ $\phi _{i}, g\in C\left(\left[\vartheta _{0}, \infty \right), \mathbb{R} \right),$ $g\left(\vartheta \right) > 0, \ \phi _{i}\geq 0$ and $\phi _{i}$ is not identically zero for large $\vartheta, \ i = 1, 2, ...\eta$;

${\bf{(M5)}}$ there exists a constant $\varepsilon \in \left(0, 1\right)$ such that

By a solution of (1.1), we mean a nontrivial function $x\in C\left(\left[\vartheta _{x}, \infty \right), \mathbb{R} \right)$, $\vartheta _{x}\geq \vartheta _{0},$ which has the properties $\upsilon \left(\vartheta \right) \in C^{3}\left(\left[\vartheta _{x}, \infty \right), \mathbb{R} \right),$ $b\left(\vartheta \right) \left(\upsilon ^{\prime \prime \prime }\left(\vartheta \right) \right) ^{\alpha }\in C^{1}\left(\left[\vartheta _{x}, \infty \right), \mathbb{R} \right)$ and $x$ satisfies (1.1) on $\left[\vartheta _{x}, \infty \right).$ We focus in our study on the solutions that satisfy $\sup \left\{ \left\vert x\left(\vartheta \right) \right\vert :\vartheta \geq \vartheta _{a}\right\} > 0$, for every $\vartheta _{a}\geq \vartheta _{x}$, and we assume that (1.1) possesses such solutions. Such a solution of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. An equation is called oscillatory if all of its solutions are oscillatory.

We frequently see repetitive movements used to illustrate different mechanical actions that occur in nature. In other words, oscillations abound in our universe. Examples of oscillatory motions include a ship moving up and down on the waves and a pendulum swing. Finding new necessary conditions for the oscillation or nonoscillation of the solutions of neutral functional differential equations, which is a component of so-called dynamical systems, has become increasingly important in recent years. As far as physicists and engineers are concerned, understanding and managing oscillations in various systems is their primary objective. Oscillation is a phenomenon that is observed in a variety of fields, including biology and economics, in addition to physics and mechanics.

One of the important differential equation branching problems is the oscillatory behavior of ordinary differential equations. The oscillatory problems to the wings of the plane can be modeled by the oscillatory problems of ordinary differential equations. Delay differential equations with deviating arguments introduce an additional layer of complexity by incorporating time delays into the modeling process. Unlike ordinary differential equations, these equations account for the influence of both current and past values of variables. In fact, differential equations with deviating arguments are widely used in physics, engineering, biology, economics, and more, making them an indispensable tool for understanding and predicting the behavior of complex phenomena, see ^{[8,9,13,15,21,23,26,27]}).

Bills and Schoenberg ^[12] examined certain oscillatory outcomes for a self-adjoint system of second-order equations. The oscillatory behavior of solutions to various classes of differential equations with a linear neutral term has been extensively studied in recent decades. However, there are few results about the oscillation of differential equations with nonlinear neutral terms; see, for example, Agarwal et al. and Grace and Graef ^[3,14,16].

Many researchers have studied the oscillatory properties of even-order differential equations on a larger scale than their odd-order counterparts. Different techniques and methods have been used to study the oscillation of different types of even-order differential equations. Illustrative and additional information can be found in references ^{[4,11,17,18,19,20,25]}. Specifically, we provide some detail.

Bazighifan and Ahmad ^[10] investigated the qualitative behavior of an even-order advanced differential equation

where $\phi _{i}\left(\vartheta \right) \geq 0, \ \delta _{i}\left(\vartheta \right) \geq \vartheta,$ and $f\left(x\right) /x^{\beta }\geq k > 0$ for $x\neq 0.$ They established sufficient conditions for oscillation of (1.5) by utilizing the theory of comparison with first-order and second-order delay equations, as well as the Riccati substitution technique.

The oscillatory behavior of the differential equations

was taken into consideration by Abdelnaser et al. ^[1]. By introducing a new set of criteria, the researchers were able to prove that all solutions to Eq (1.6) oscillate.

Muhib et al. ^[22] considered a class of neutral differential equations

They used Riccati transformations to present new conditions for the oscillation of (1.7), where $\rho \left(\vartheta \right) \geq 0, \ g\left(\vartheta \right) \geq 0, \ \gamma$ and $\hbar$ are ratios of odd positive integers with $\gamma \geq 1$, $0 < \hbar < 1,$ and $f\in C\left(\left[\vartheta _{0}, \infty \right) \times \mathbb{R}, \mathbb{R} \right)$ and there exists $\phi \in C\left(\left[\vartheta _{0}, \infty \right), \left(0, \infty \right) \right)$ such that $\left\vert f\left(\vartheta, x\right) \right\vert \geq \phi \left(\vartheta \right) \left\vert x\right\vert ^{\beta }.$ Below, we present one of the results in ^[22].

Theorem 1.1. Assume that

holds. If

for all $\lambda \in \left(0, 1\right)$ and $N > 0$, then (1.7) is oscillatory, where

and

for some $\epsilon \in \left(0, 1\right)$ and$\ k_{1}, k_{2} > 0$.

Agarwal et al. ^[2] studied the oscillation of a neutral differential equation

where $n$ is even and $g\left(\vartheta \right) \geq 0.$ They established some sufficient conditions for oscillation of (1.9) using the Riccati transformation technique. Below, we present one of the results in ^[2].

Theorem 1.2. Let $n\geq 4$ be even (M3), (M4), and

hold. If the equation

is oscillatory for some constant $\lambda _{0}\in \left(0, 1\right)$, and the equation

is oscillatory, then (1.9) is oscillatory, where

and

The objective of this paper is to provide new results of oscillation (1.1) in canonical form, which would improve and extend some previous literature. In addition, an example is given that shows the applicability of the results we obtained.

The following notation will be used in the remaining sections of this work:

2.   Preliminary lemmas

For the proof of our main results, we need to give the following lemmas:

Lemma 2.1. ^[5] Let $f\in C^{n}\left(\left[\vartheta _{0}, \infty \right), \left(0, \infty \right) \right)$, the derivative $f^{\left(n\right) }\left(\vartheta \right)$ is of fixed sign and not identically zero on a subray of $\left[\vartheta _{0}, \infty \right),$ and there exists a $\vartheta _{x}\geq \vartheta _{0}$ such that $f^{\left(n-1\right) }\left(\vartheta \right) f^{\left(n\right) }\left(\vartheta \right) \leq 0$ for all $\vartheta \geq \vartheta _{1}$. If $\lim_{\vartheta \rightarrow \infty }f\left(\vartheta \right) \neq 0$, then for every $\lambda \in \left(0, 1\right)$ there exists a $\vartheta _{\lambda }\geq \vartheta _{1}\ $ such that

for all $\vartheta \geq \vartheta _{\lambda }.$

Lemma 2.2. [7, Lemma 1] Let the function $f$ satisfy $f^{\left(n\right) }\left(\vartheta \right) > 0, \ n = 1, 2, ..., \kappa,$ and $f^{\left(\kappa +1\right) }\left(\vartheta \right) \leq 0$ eventually. Then, for every $\varepsilon \in \left(0, 1\right),$

eventually.

Lemma 2.3. ^[6] Let $x\ $be a positive solution of (1.1), and (1.3) holds. Then, $\left(b\left(\vartheta \right) \left(\upsilon ^{\prime \prime \prime }\left(\vartheta \right) \right) ^{\alpha }\right) ^{\prime } < 0,$ we also find that there exist two potential cases eventually, which are as follows:

Lemma 2.4. ^[24] If $y$ is a positive and strictly decreasing solution of the integral inequality

then there exists a positive solution $x\left(\vartheta \right)$ of the differential equation

being such that $x\left(\vartheta \right) \leq y\left(\vartheta \right)$ for all large $\vartheta$ and satisfying $\lim_{\vartheta \rightarrow \infty }x^{\left(i\right) }\left(\vartheta \right) = 0$ monotonically $\left(i = 1, 2, ..., n-1\right),$ where $f$ is a continuous function defined on $\left[\vartheta _{0}, \infty \right) \times \left[0, \infty \right) ^{m}$ and $g_{j}\left(\vartheta \right)$ are continuous functions on the interval $\left[\vartheta _{0}, \infty \right)$ such that

The function $f = f\left(\vartheta, u_{1}, u_{2}, ..., u_{m}\right)$ is assumed to be increasing in each of $u_{1}, u_{2}, ..., u_{m}$. Moreover, it is supposed that $f$ is positive on $\left[\vartheta _{0}, \infty \right) \times \left[0, \infty \right) ^{m}$ and that the delays $\vartheta -g_{j}\left(\vartheta \right)$ are positive for $\vartheta \geq \vartheta _{0}$, i.e., $g_{j}\left(\vartheta \right) < \vartheta$ for every $\vartheta \geq \vartheta _{0}$ and$\ \left(j = 1, 2, ..., m\right)$.

3.   Main results

We now present the main results of this paper.

Theorem 3.1. Assume that $\beta \geq 1\ $and that there exists a positive function $\mu \in C^{1}\left(\left[\vartheta _{0}, \infty \right), \mathbb{R} \right)$ such that

If

and

have no positive solutions, then every solution of (1.1) is oscillatory, where $q\left(\vartheta \right) = \tau ^{-1}\left(\mu \left(\vartheta \right) \right),$ $c > 0$ and $\epsilon _{1}, \epsilon _{2}\in \left(0, 1\right)$.

Proof. Assume that Eq (1.1) has a nonoscillatory solution $x\left(\vartheta \right)$, say $x\left(\vartheta \right) > 0$, $x\left(\delta \left(\vartheta \right) \right) > 0$, and $x\left(\tau \left(\vartheta \right) \right) > 0$ for $\vartheta \geq \vartheta _{1}\geq \vartheta _{0}$. From (1.2), we find

and so

We first consider what Case (1) holds. Since $\kappa = 3$, in view of (2.2), for every $\varepsilon \in \left(0, 1\right),$ we get

now,

using (3.5), we find

Since $\tau \left(\vartheta \right) \leq \vartheta$ and $\tau ^{\prime }\left(\vartheta \right) > 0$, $\left(\tau ^{-1}\left(\vartheta \right) \right) ^{\prime } > 0$ and furthermore $\vartheta \leq \tau ^{-1}\left(\vartheta \right)$. Thus,

By using (3.7) and (3.8), it follows that

and so

From (3.4) and (3.9), we find

From (M5), there exists a $\epsilon _{1}\in \left(0, 1\right)$ such that

Using the above inequality in (3.10) gives

From (1.1) and (3.11), we have

and so

In view of the fact that $\mu \left(\vartheta \right) \leq \delta \left(\vartheta \right)$ and $\upsilon ^{\prime }\left(\vartheta \right) > 0$, inequality (3.12) becomes

Since $\upsilon \left(\vartheta \right) > 0$ and $\upsilon ^{\prime }\left(\vartheta \right) > 0$, there exists a constant $c > 0$ such that

From (3.13), (3.14), and $\beta \geq 1$, we find the following differential inequality:

has a positive solution $\upsilon.$ This implies that (3.2) also has a positive solution, which contradicts our assumption.

Now, we consider what Case (2) holds. Since $\kappa = 1$, in view of (2.2), for every $\varepsilon \in \left(0, 1\right),$ we get

from which we see that

By (3.8) and (3.17),

Combining (3.4) and (3.18), we obtain

From (M5), there exists a $\epsilon _{2}\in \left(0, 1\right)$ such that

and using this in (3.19) implies

Using (3.20) in (1.1) yields

and so

Since $\mu \left(\vartheta \right) \leq \delta \left(\vartheta \right)$ and $\upsilon ^{\prime }\left(\vartheta \right) > 0$, then (3.21) becomes

In view of (3.14) and $\beta \geq 1$, we find the following differential inequality:

has a positive solution $\upsilon.$ This implies that (3.3) also has a positive solution, which contradicts our assumption. The proof is now complete. □

Theorem 3.2. Assume that $\beta < 1$ and there exists a positive function $\mu \in C^{1}\left(\left[\vartheta _{0}, \infty \right), \mathbb{R} \right)$ such that (3.1) holds. If

and

have no positive solutions, then every solution of (1.1) is oscillatory, where $q\left(\vartheta \right) = \tau ^{-1}\left(\mu \left(\vartheta \right) \right),$ $d_{1}, d_{2} > 0$ and $\epsilon _{1}, \epsilon _{2}\in \left(0, 1\right).$

Proof. Assume that Eq (1.1) has a nonoscillatory solution $x\left(\vartheta \right)$, say $x\left(\vartheta \right) > 0$, $x\left(\delta \left(\vartheta \right) \right) > 0$, and $x\left(\tau \left(\vartheta \right) \right) > 0$ for $\vartheta \geq \vartheta _{1}\geq \vartheta _{0}$.

We first consider what Case (1) holds. By performing the same steps as in the proof of Theorem 3.1, we arrive at (3.7) and (3.13). By (3.7), there exists a constant $d_{1} > 0$ such that

and so

Using (3.26) in (3.13) and applying the fact that $\beta < 1$ yields

That is, (3.24) has a positive solution, a contradiction.

Now, we consider what Case (2) holds. By performing the same steps as in the proof of Theorem 3.1, we arrive at (3.17) and (3.22). By (3.17), there exists a constant $d_{2} > 0$ such that

and so

Using (3.28) in (3.22) and applying the fact that $\beta < 1$ yields

That is, (3.25) has a positive solution, a contradiction. The proof is now complete. □

Theorem 3.3. Assume that $\beta \geq 1$ and there exists a positive function $\mu \in C^{1}\left(\left[\vartheta _{0}, \infty \right), \mathbb{R} \right)$ such that (3.1) holds. If

and

are oscillatory, for some constants $\lambda, \varepsilon \in \left(0, 1\right)$, then every solution of (1.1) is oscillatory, where $q\left(\vartheta \right) = \tau ^{-1}\left(\mu \left(\vartheta \right) \right),$ $c > 0$ and $\epsilon _{1}, \epsilon _{2}\in \left(0, 1\right).$

Proof. Assume that Eq (1.1) has a nonoscillatory solution $x\left(\vartheta \right)$, say $x\left(\vartheta \right) > 0$, $x\left(\delta \left(\vartheta \right) \right) > 0$, and $x\left(\tau \left(\vartheta \right) \right) > 0$ for $\vartheta \geq \vartheta _{1}\geq \vartheta _{0}$.

We first consider what Case (1) holds. By performing the same steps as in the proof of Theorem 3.1, we arrive at (3.15). Since $\upsilon \left(\vartheta \right) > 0$ and $\upsilon ^{\prime }\left(\vartheta \right) > 0$, we have lim$_{\vartheta \rightarrow \infty }\ \upsilon \left(\vartheta \right) \neq 0$. Thus, by Lemma 2.1, we obtain

from which we see that

Using (3.33) in (3.15) yields

With $y\left(\vartheta \right) = b\left(\vartheta \right) \left(\upsilon ^{\prime \prime \prime }\left(\vartheta \right) \right) ^{\alpha }$, we find $y\left(\vartheta \right)$ is a positive solution of the differential inequality

It follows from Lemma 2.4 that the differential equation (3.30) also has a positive solution for all $\lambda \in \left(0, 1\right)$, but this contradicts our assumption on (3.30).

Now, we consider what Case (2) holds. By performing the same steps as in the proof of Theorem 3.1, we arrive at (3.16) and (3.23). Integrating (3.23) from $\vartheta$ to $\infty$ gives

and so

Integrating (3.35) from $\vartheta$ to $\infty$, we have

and so

Using (3.16) in (3.36) yields

With $\omega \left(\vartheta \right) = \upsilon ^{\prime }\left(\vartheta \right)$, we see that $\omega \left(\vartheta \right)$ is a positive solution of the differential inequality

for every $\varepsilon \in \left(0, 1\right)$. We finalize the proof using the same method as outlined in Case (1). The proof is now complete. □

Corollary 3.1. Let $\alpha = 1$ and $\beta \geq 1$ hold. Assume further that there exists a positive function $\mu \in C^{1}\left(\left[\vartheta _{0}, \infty \right), \mathbb{R} \right)$ such that (3.1) holds. If

and

then every solution of (1.1) is oscillatory, where $q\left(\vartheta \right) = \tau ^{-1}\left(\mu \left(\vartheta \right) \right).$

Proof. We first consider what Case (1) holds. By performing the same steps as in the proof of Theorem 3.3, we arrive at (3.34). Integrating (3.34) from $q\left(\vartheta \right)$ to $\vartheta$ and then using $\alpha = 1$ and the fact that $y$ is a decreasing function, we see that

and so

this can be expressed as follows:

so this contradicts (3.39).

Now, we consider what Case (2) holds. By performing the same steps as in the proof of Theorem 3.3, we arrive at (3.38). Integrating (3.38) from $q\left(\vartheta \right)$ to $\vartheta$ and then using $\alpha = 1$ and the fact that $\omega$ is a decreasing function, we see that

and so

this can be expressed as follows:

which contradicts (3.40). The proof is now complete. □

Theorem 3.4. Assume that $\beta < 1$ and there exists a positive function $\mu \in C^{1}\left(\left[\vartheta _{0}, \infty \right), \mathbb{R} \right)$ such that (3.1) holds. If

and

are oscillatory for some constants $\lambda, \varepsilon \in \left(0, 1\right),$ then every solution of (1.1) is oscillatory, where $q\left(\vartheta \right) = \tau ^{-1}\left(\mu \left(\vartheta \right) \right),$ $d_{1}, d_{2} > 0$ and $\epsilon _{1}, \epsilon _{2}\in \left(0, 1\right).$

Proof. Assume that Eq (1.1) has a nonoscillatory solution $x\left(\vartheta \right)$, say $x\left(\vartheta \right) > 0$, $x\left(\delta \left(\vartheta \right) \right) > 0$, and $x\left(\tau \left(\vartheta \right) \right) > 0$ for $\vartheta \geq \vartheta _{1}\geq \vartheta _{0}$.

We first consider what Case (1) holds. By performing the same steps as in the proof of Theorem 3.2, we arrive at (3.27). Since $\upsilon \left(\vartheta \right) > 0$ and $\upsilon ^{\prime }\left(\vartheta \right) > 0$, we have $\lim_{\vartheta \rightarrow \infty }\upsilon \left(\vartheta \right) \neq 0$ and so by Lemma 2.1, we find (3.32) holds. Using (3.32) in (3.27) gives

With $y\left(\vartheta \right) = b\left(\vartheta \right) \left(\upsilon ^{\prime \prime \prime }\left(\vartheta \right) \right) ^{\alpha }$, we see that $y\left(\vartheta \right)$ is a positive solution of the differential inequality

It follows from Lemma 2.4 that the differential equation (3.41) also has a positive solution for all $\lambda _{1}\in \left(0, 1\right)$, but this contradicts our assumption on (3.41).

Now, we consider what Case (2) holds. Then again (3.16) holds for every $\varepsilon \in \left(0, 1\right)$. By performing the same steps as in the proof of Theorem 3.2, we arrive at (3.29). Integrating (3.29) from $\vartheta$ to $\infty$, we obtain

and so

Integrating (3.44) from $\vartheta$ to $\infty$, we obtain

and so

With $\omega \left(\vartheta \right) = \upsilon ^{\prime }\left(\vartheta \right)$ and using (3.16) in (3.45), we see that $\omega \left(\vartheta \right)$ is a positive solution of

We finalize the proof using the same method as outlined in Case (1). The proof is now complete. □

Corollary 3.2. Let $\alpha = 1$ and $\beta < 1$ hold. Assume further that there exists a positive function $\mu \in C^{1}\left(\left[\vartheta _{0}, \infty \right), \mathbb{R} \right)$ such that (3.1) holds. If

and

then every solution of (1.1) is oscillatory, where $q\left(\vartheta \right) = \tau ^{-1}\left(\mu \left(\vartheta \right) \right).$

Proof. The details of the proof are omitted as they are similar to those of Corollary 3.1. □

We provide the following example to demonstrate our results:

Example 3.1. Consider the neutral differential equation

It is easy to verify that

Choosing $\mu \left(\vartheta \right) = \vartheta /A_{4}\ $and $A = \max \left\{ A_{2}, A_{3}\right\},$ where $A_{4} > A_{1},$ $A_{4}\geq A$ and $A_{1}, A_{2}, A_{3}, A_{4} > 1,$ then (3.1) holds. We also find that

Let $\varepsilon = 1/4;$ we see that (1.4) holds.

Now, it is easy to check that the condition (3.39) is satisfied, where

Moreover, we find that condition (3.40) is satisfied, where

Thus, using Corollary 3.1, every solution of (3.48) is oscillatory.

Remark 3.1. Consider the neutral differential equation

as a special case of Eq (3.48), we see that Theorem 1.1 cannot be applied to (3.49) since

accordingly, Theorem 1.1 fails to study the oscillation of (3.49).

Also, we see that Theorem 1.2 cannot be applied to (3.49) since $\delta \left(\vartheta \right) = \vartheta /2$ is greater than $\tau \left(\vartheta \right) = \vartheta /3$ for $\vartheta \geq 1$. Accordingly, Theorem 1.2 fails to study the oscillation of (3.49).

Now, by using Corollary 3.1 and choosing $\mu \left(\vartheta \right) = \vartheta /4$ and $\varepsilon = 1/4$, we see that the conditions (3.39) and (3.40) are satisfied and therefore, Eq (3.49) is oscillatory.

4.   Conclusions

By using comparison principles, we analyze the asymptotic behavior of solutions to a class of fourth-order neutral differential equations. We have obtained some new oscillation results for (1.1) in the case where $\int_{\vartheta }^{\infty }1/b^{1/\alpha }\left(\zeta \right) \mathrm{d} \zeta = \infty$. These results ensure that all solutions to the studied equation are oscillatory, and they also improve and extend some results from previous studies. It will be of interest to investigate the higher-order differential equations of the form

where $n\geq 4$ is an even natural number.

Author contributions

The authors contributed equally to this work. Both of the authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

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

Acknowledgments

The authors acknowledge the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia, under project Grant No. 6018.

Conflict of interest

The authors declare no conflicts of interest.