Dynamics of a general model of nonlinear difference equations and its applications to LPA model

1.   Introduction

In phenomena where the plurality of observations of a temporally changing variable are discrete, difference equations (DE) are utilized to explain how a phenomenon develops in real life. These equations therefore become crucial in mathematical models. Applications involving higher-order nonlinear DEs are very common. Furthermore, the DEs naturally arise as numerical solutions and discrete analogs to ordinary and functional differential equations that model a wide range of distinct biological, physical, and economic events, see ^[1,2,3,4,5].

The global asymptotic behavior, as a qualitative characteristic of solutions to linear DEs, has significant applications in a variety of fields, such as control theory, biology, neural networks, and many more. It is difficult to verify the global stability of solutions to a DE using numerical methods. As a result, many mathematicians and engineers are interested in the analytical study of those qualitative features because it is the only way to understand those properties (see ^{[6,7,8,9,10,11,12]}).

In ^[13], Sun and Xi investigated the stability of the DE

where $k$ and $l$ are in $\mathbb{Z}$, $k < l$, and $F\left(s, t\right)$ is non-increasing in $s$ and non-decreasing in $t$. In ^[14], Kocic et al. studied the stability of

which describes the dynamics of baleen whales.

The study of general models of difference equations, despite its difficulty compared to the study of specific models, provides more results, creates new methods, and gives a more general overview of similar models. Abdelrahman et al. ^[15,16] investigated the qualitative properties of the DE

where $\alpha, \beta \in \left[ 0, \infty \right)$, $F\in C\left(\left(0, \infty \right) ^{2}, \left(0, \infty \right) \right)$, and $F$ is homogenous with degree $0$.

Recently, Moaaz et al. ^[17,18,19] examined the asymptotic features of the DEs

and

Definition 1. ^[20] The function $\phi$ is called a homothetic function if there exist functions $G: \mathbb{R} \rightarrow \mathbb{R}$ and $H: \mathbb{R} ^{2}\rightarrow \mathbb{R}$, such that $G$ is a monotonic function, $H$ is a homogenous function with degree $\kappa$, and $\phi \left(t, s\right) = G\left(H\left(t, s\right) \right).$

In this work, we study the general DE

where $\Phi$ is a homothetic function, and

(H1) $m, \ell \in \mathbb{Z} ^{+}$, $\alpha \in \left[ 0, \infty \right)$, $\beta,$ $B_{-\ell },$ $B_{-\ell +1},$ $...,$ $B_{0}\in \left(0, \infty \right)$.

(H2) $\Phi$ is a homothetic function, where $\Phi \left(t, s\right) = G\left(H\left(t, s\right) \right)$ and $H$ is a homogenous function with degree $\kappa > 0$.

Theorem 1. [21, Theorem 1.3.1] Suppose that $f_{1},$ $f_{2}\in C\left(\left[ 0, \infty \right) ^{2}, \left[ 0, 1\right) \right)$ satisfy

(ⅰ) $f_{1}$ and $f_{2}$ are non-increasing in each of their arguments$;$

(ⅱ) $f_{1}\left(u, u\right) > 0$ for $u\geq 0;$

(ⅲ) $f_{1}\left(u, v\right) +f_{2}\left(u, v\right) < 1$ for $u, v\in \left(0, \infty \right)$.

Then the zero equilibrium of the DE

is globally asymptotically stable.

2.   Behavior of solutions of (E)

Here, we define $\mathcal{F}:\left[ 0, \infty \right) \times \left[ 0, \infty \right) \rightarrow \left(0, \infty \right)$ by

A fixed point $p^{\ast }$, in $\left[ 0, \infty \right) \times \left[ 0, \infty \right)$, is called an equilibrium point (EP) of the DE $B_{m+1} = \mathcal{F}\left(B_{m}, B_{m-\ell }\right)$. The study of any physical system's dynamics centers on the concept of EPs (states). All states (solutions) of a particular system tend to its equilibrium state, which is known from multiple applications in science.

Now, we assume $\mathcal{F}\left(p^{\ast }, p^{\ast }\right) = p^{\ast }$ and look for the positive EPs. Then,

Thus, we obtain

Since $\Phi = G\left(H\right)$ and $H$ is a homogenous function with degree $\kappa$, we obtain

Hence,

under the conditions $\kappa \neq 0,$ $H\left(1, 1\right) > 0$ and

Thus, EPs of (E) are $zero$ and the positive value defined in (2.3).

2.1.   Stability of EPs

This section is concerned with determining the criteria that guarantee that the EP of DE (E) is locally asymptotically stable (LAS), or globally asymptotically stable (GAS).

Determining the behavior of solutions of DEs near EPs is one of the main objectives when studying them. It is easy to verify that

and

By substituting for $\left(t, s\right) = \left(0, 0\right)$ in (2.5) and (2.6), we have that the linearized equation about the trivial EP is

When $\ell = 1$, employing [21, Theorem 1.1], we get that the trivial EP of Eq (E) is (see Figure 1):

(a) LAS and sink if $\alpha < 1-\beta \, G\left(0\right) < 2$;

(b) unstable and repeller if $\alpha > 1$ and $\alpha < \left\vert 1-\beta \, G\left(0\right) \right\vert$;

(c) saddle point if $\alpha +4\beta \, G\left(0\right) > 0$ and $\alpha > \left\vert 1-\beta \, G\left(0\right) \right\vert$;

(d) nonhyperbolic point if $\alpha = \left\vert 1-\beta \, G\left(0\right) \right\vert$.

From Figure 1, it is easy to see the areas where the different types of stability and instability occur. We also notice that LAS occurs when the values of $\alpha$ and $\widehat{\beta }: = \beta G\left(0\right)$ lie below the straight line $\alpha +\widehat{\beta } = 1$.

Whereas when $\ell > 1$, the trivial EP of Eq (E) is LAS if $\alpha +\beta \left\vert G\left(0\right) \right\vert < 1$. On the other hand, the linearized equation about the positive EP is

where

and

By using the Clark criterion (see [14, Theorem 1.3.7]), we get that the positive EP of (E) is asymptotically stable if

Next, we study the global stability of EPs of Eq (E).

Theorem 2. Assume that $\ell = 1,$ $\Phi$ has non-positive partial derivatives, and there is a real number $c$ such that $0 < \Phi \left(t, s\right) \leq c$ for all $t, s\in \left[ 0, \infty \right)$. Then the $zero$ EP of (E) is GAS if $\alpha > 0$ and $\alpha +c\beta < 1$.

Proof. Assume that $\mathcal{F}$ is defined as in (2.1). Now, we define the functions

Then, Eq (E) takes the form

Hence, it is easy to notice the following:

1) $f_{0}$ and $f_{1}$ are non-increasing with respect to $t$ and $s$.

2) $f_{0}\left(t, t\right) > 0$ for all $t\geq 0$.

3) Since $\alpha +c\beta < 1$, we have

and so $\alpha +\beta \, \Phi \left(t, s\right) < 1$. Therefore,

From Theorem 1, the $zero$ EP of (E) is GAS. This completes the proof.

Theorem 3. Suppose that $\alpha \in \left(0, 1\right)$ and there is a real number $h$ such that $0 < \Phi \left(t, s\right) \leq h$. If $\alpha +\beta h < 1$, then $\lim_{m\rightarrow \infty }B_{m} = 0$.

Proof. We define the sequence $\left\{ \widetilde{B}_{m}\right\} _{m = 0}^{\infty }$ by

From (E), we see that

Hence, the sequence $\left\{ \widetilde{B}_{m}\right\} _{m = 0}^{\infty }$ is nonincreasing, and so $\lim_{t\rightarrow \infty }\widetilde{B} _{m} = B_{0}\geq 0$.

Suppose that $B_{0} > 0$. Then, for all $\epsilon > 0$, there is a $M > 0$ such that $B_{m}\leq \widetilde{B}_{m} < B_{0}+\epsilon$ for all $m > M$. It follows from (2.10) that

Since $\alpha +\beta h < 1$, there is a $\epsilon > 0$ small enough such that $\alpha +\beta h < 1-2\epsilon$, and so

which with (2.11) gives $B_{m+1} < B_{0}-\epsilon$. Therefore, $\lim_{t\rightarrow \infty }B_{m}\leq B_{0}-\epsilon$, a contradiction. Then, $B_{0} = 0$ and $\lim_{t\rightarrow \infty }B_{m} = 0$. This completes the proof.

Lemma 1. Suppose that $\Phi ^{\prime }\left(t, t\right) < 0$. Then, $\mathcal{ F}\left(t, t\right)$ satisfies the negative feedback condition

for $t\in \mathbb{R} ^{+}-\left\{ p^{\ast }\right\}$.

Proof. We define the function

Then,

It is easy to notice that $\Theta \left(t\right) = 0$ if and only if $t = 0$ or $p^{\ast }$. Now, assume $t < p^{\ast }$. Hence, $\Phi \left(t, t\right) > \Phi \left(p^{\ast }, p^{\ast }\right) = (1-\alpha)/{\beta },$ and then $\Theta \left(t\right) > 0$. Also, if $t > p^{\ast },$ then $\Theta \left(t\right) < 0$. Therefore, $\mathcal{F}\left(t, t\right)$ satisfies condition (2.12). This completes the proof.

Lemma 2. Suppose that $\alpha \in \left(0, 1\right)$, $\lambda > 0$, and there is a real number $c$ such that $s\Phi \left(t, s\right) \leq c$. Then

Proof. From (E), we have

It is easy to verify that the sequence on the right side has the solution

Then

Taking $\lim \sup$ as $m\rightarrow \infty$, we arrive at (2.13). The proof is complete.

2.2.   Existence of periodic solutions

We create, in the following, a criterion to ensure that there are two-cycle periodic solutions to Eq (E).

Theorem 4. Suppose that $\ell$ is odd. DE (E) has a prime period two solution if and only if there is a real number $\gamma > 0$ such that

and this solution is $\left\{ S_{m}\right\} _{m = -2}^{\infty }$ where

Proof. Suppose that (E) has a prime period two solution $\left\{..., l, k, l, k, l, k, ...\right\}$. It follows from (E) that

From (2.15), we obtain

and so,

Similarly, from (2.16), we obtain

From (2.17) and (2.18), we arrive at (2.14).

Conversely, we assume that (2.14) holds. Now, we choose

for $\gamma \in \mathbb{R} ^{+}$. Hence, from (2.14), we find

Then,

Similarly, we have that $B_{2} = B_{0}$. Therefore, Eq (E) has a prime period two solution.

Theorem 5. Suppose that $\ell$ is even. DE (E) has a prime period two solution if and only if there exists a real number $\gamma > 0$ such that

and this solution is $\left\{ S_{m}\right\} _{m = -2}^{\infty }$ where

Proof. Suppose that (E) has a prime period two solution $\left\{..., l, k, l, k, l, k, ...\right\}$. It follows from (E) that

From (2.20), we obtain

and then

Similarly, from (2.21), we obtain

Combining (2.17) and (2.18), we arrive at (2.19).

We omitted the rest of the proof because it is similar to the proof of the previous theorem.

Remark 1. Equations (1.2)–(1.5) are special cases of Eq (E). Therefore, we can obtain the results for stability and periodicity in ^{[15,16,17,18,19]} by imposing different forms for the function $\Phi$ and the parameters $\alpha$ and $\beta$.

3.   The flour beetle model

The genus Tribolium or Tenebrio of darkling beetles includes flour beetles. They are common laboratory animals because they are easy to keep. The flour beetles eat wheat and other grains, can endure even more radiation than cockroaches, and are designed to live in very arid settings ^[22]. They are a significant pest in the agriculture sector and have a high level of pesticide resistance. These insects are present all over the world and can infest food that has been preserved, which can alter the flavor of the food.

The larva, pupa, and adult stages of the flour beetle's life cycle are separated by approximately two weeks. The species also shows nonlinear interactions between life stages, such as moving stages cannibalizing non-moving stages. Population dynamics are made possible, which is highly interesting.

The "Larvae-Pupae-Adult" (LPA) model, which describes the dynamics of flour beetle population dynamics, is one of the most thoroughly studied and well-validated models in mathematical ecology. The following system of three DEs provides the model:

where $L_{m}$, $P_{m}$, and $A_{m}$ are the number of larvae at time $t$, the number of individuals in the "$P$ stage" (including non-feeding larvae, pupae, and callow adults) at time $t$, and the number of sexually mature adults at time $t$, respectively. $k$ is the rate at which an adult lays eggs. $\mu _{ea}$ and $\mu _{el}$ are cannibalism coefficients of eggs by larvae and eggs by adults, respectively. Pupae must escape cannibalism by adults $\mu _{pa}$ to become adults. $\eta _{a}$ and $\eta _{l}$ are the rates of adult death and naturally larval death, respectively.

In the simplified case when larval cannibalism of eggs is not present, i.e., $\mu _{el} = 0$, we observe that the DE

represents system (3.1), where $\eta _{a},$ $\eta _{l}\in \left(0, 1\right),$ $k > 0,$ $\mu _{pa}+\mu _{el} > 0$, and

Now, to apply the results of the previous section, we set that

Then, we conclude that the EPs of model (3.2) are $p^{\ast } = 0$ or

which is positive if $k\left(1-\eta _{l}\right) > \eta _{a}$. Moreover, the stability behavior of EPs is as follows:

- The trivial EP of (3.2) is LAS if $\eta _{a} > k\left(1-\eta _{l}\right)$.

- The positive EP of (3.2) is LAS if

where

- There exists a constant $c = 1/\left(\mathrm{e}\mu _{ea}\right)$ such that $s\Phi \left(t, s\right) \leq c$, and then

- There exists a constant $h = 1$ such that $\Phi \left(t, s\right) \leq h$. Then $\lim_{m\rightarrow \infty }B_{m} = 0$, if $\eta _{a} > k\left(1-\eta _{l}\right)$.

Figure 2 shows the region where local stability occurs for the zero EP of model (3.2). We note that the relationship between the ratio $\frac{\mu _{a}}{1-\mu _{l}}$ and $k$ (the number of eggs laid per adult per unit of time) is the one that controls the stability or instability of the zero EP of model (3.2). We notice that with the increase in rates of adult death and naturally larval death (approaching 1), the system approaches stability, and in the case of a decrease (approaching 0) of these rates, the system turns into instability. As for the instability points, they are of the saddle points if the ratio $\frac{\mu _{a}}{1-\mu _{l}}$ is less than the number of eggs laid per adult per unit of time, and of the nonhyperbolic points if this ratio is equal to $k$.

Lemma 3. Assume that $\mathcal{F} \left(t, s\right) = \alpha t+\beta s\exp \left(-c_{1}s-c_{2}t\right)$. If $\beta < \mathrm{e}\left(1-\alpha \right)$ and $\beta \, c_{2} < \mathrm{e}\, \alpha \, c_{1}$, then $\mathcal{F} _{t}\ $ and $\mathcal{F} _{s}$ are non-negative.

Proof. From (2.5), we have $\mathcal{F} _{t}\geq \alpha -c_{2}\beta \, s\exp \left(-c_{1}s\right) \geq \alpha -\frac{c_{2}\beta }{ c_{1}\mathrm{e}}$, which with the fact that $\beta \, c_{2} < \mathrm{e} \, \alpha \, c_{1}$ gives $\mathcal{F} _{t}\geq 0$. Also, we have that $\mathcal{F} _{s} = \beta \left(1\, -c_{1}s\right) \exp \left(-c_{1}s-c_{2}t\right)$, which is positive by using (3.4) and the fact that $\beta < \mathrm{e}\left(1-\alpha \right)$. The proof is complete.

Theorem 6. If

then the positive EP of (3.2) is GAS.

Proof. It is easy to notice that Lemmas 1 and 3 guarantee the conditions of global stability in Theorem 1.4.1 in ^[21].

Corollary 1. Model (3.2) has a prime period two solution if and only if there exists a positive real number $\gamma$ such that

and this solution is $\left\{ S_{m}\right\} _{m = -2}^{\infty }$ where

Remark 2. Kuang and Cushing ^[23] and Brozak et al. ^[24] presented the global stability results for model (3.1). The previous global stability results agree with the results presented in ^[23,24] but the results in ^[23,24] do not provide any information about the periodicity of the solutions of model (3.1).

4.   Numerical simulation examples

Here, we provide some numerical examples that support the results in the previous sections.

Example 1. Consider the Baleen Whales model

where $\beta \in \left(0, 1\right)$, and $\eta, \, M\in \left(0, \infty \right)$. Note that $\Phi \left(t, s\right) = G\left(H\left(t, s\right) \right),$ where

The positive EP of (4.1) $p^{\ast } = M$ is asymptotically stable if $\kappa \eta < 2$, see Figure 3a. We notice in this example that the EP is not affected by the parameters $\kappa$, $\beta$, and $\eta$, which means that the EP will remain constant no matter how the values of these parameters change, as is clear from Figure 3a. In Figure 3b, we note that Eq (4.1) has the prime period two solution $\left\{..., 6, 12, 6, 12, ...\right\}$ when $\kappa = 1,$ $\ell = 2,$ $\beta = 0.5,$ $\eta = 5,$ and $M = 10$, as Theorem 5 indicates. We also notice that any slight change in the initial conditions causes a defect in the periodic solution of the equation.

Example 2. Consider the DE

where $\beta, a\in \left(0, \infty \right)$ and $\alpha, \, B_{-\ell }, \, B_{-\ell +1}, \, ...\, ,B_{0}\in \left[ 0, \infty \right)$. Note that $\Phi \left(t, s\right) = G\left(H\left(t, s\right) \right),$ where

$Zero$ EP of Eq (4.2) is LAS if $\alpha a+\beta < a$. Moreover, it is easy to notice that $0 < \Phi \left(t, s\right) \leq \frac{1}{a}$. Then, it follows from Theorem 2 that $zero$ EP of (4.2) is GAS if $\alpha a+\beta < a$. Figure 4a shows a set of stable solutions to Eq (4.2). We notice that stability is not affected by the initial data as long as the condition $\alpha a+\beta < a$ is satisfied.

The positive EP of (4.2) is

which is LAS if

In Figure 4b, we see that some solutions to Eq (4.2) are stable with different initial data; this is consistent with condition (2.9).

Example 3. Consider the flour beetle model (3.2) when $\eta _{a} = \eta _{l} = 0.5,$ and $\mu _{ea} = \mu _{pa} = 1$, namely,

Figure 5a shows two solutions of (3.2), one of which is stable (when $k = 2$) and the other is unstable (when $k = 300$); this supports the validity of condition (3.3). Figure 5b shows a periodic solution of Eq (4.3) when $k = 100$, as Theorem 5 indicates.

5.   Conclusions

In biology, DEs are frequently employed as mathematical representations of actual biological phenomena. In this work, we examined the qualitative characteristics of solutions to a general model of DEs, with the flour beetle model serving as a particular example. We studied the periodic behavior of the solutions to this model, as well as boundedness and stability. The numerical examples that are provided validate the findings and show how the solutions for the model under study behave. As an interesting future research point, we propose to study the oscillatory and bifurcation behavior of the general model studied. We also hope in the future to be able to obtain similar results that include cases where $\kappa$ is negative, as well as to study the existence of periodic solutions with period three and the global stability of periodic solutions.

Use of AI tools declaration

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

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research and Libraries in Princess Nourah bint Abdulrahman University for funding this research work through the Research Group project, Grant No. (RG-1445-0039).

Conflict of interest

The authors declare there are no conflicts of interest.