Discrete Leslie’s model with bifurcations and control

: We explored a local stability analysis at fixed points, bifurcations, and a control in a discrete Leslie’s prey-predator model in the interior of R 2 + . More specially, it is examined that for all parameters, Leslie’s model has boundary and interior equilibria


Motivation and literature survey
Ecological balance is defined as a "state of dynamic equilibrium within an organismal community in which genetic, species, and ecosystem diversities remain relatively stable". An ecological imbalance results from a breakdown of an ecosystem's natural equilibrium. Natural habitat degradation, climate change, global warming, and biodiversity loss are all results of an ecological imbalance. Therefore, ecological balance is necessary to preserve the diverse and abundant organismal variety. The survival, reproduction, and fitness of an individual are determined by responses to environmental factors. When one creature uses a habitat or food supply in a certain way, it modifies the temporal and spatial dynamics of the habitat structure and resource distribution for another organism. To maintain the balance and stability, scientists and ecologist study different factors involved in ecology where prey-predator is most essential to consider [1]. Ecological relationships between predators and prey are crucial. Predators pursue and consume prey, which has an impact on both population's dynamics.
The population dynamics of predators and prey are shown mathematically by the Lotka-Volterra model. The model may be used to estimate how predator and prey populations can evolve under various situations by taking variables like birth rates, death rates, and the availability of resources into consideration. For instance, if there are many preys and few predators, the predator population will probably grow since more food is available. However, if the quantity of predators is too high, it may diminish the number of prey, resulting in a drop in both populations. Ecologists may learn about the numerous links that exist between various species within an ecosystem and create plans for biodiversity preservation and resource management by understanding predator-prey interactions. Prey-predator models have important mathematical implications because they clarify ecosystem dynamics and the interactions of various species. These models enable us to estimate how these populations will evolve over time by using mathematical equations to represent the interactions between predators and prey. For instance, differential equalizations are used in the popular prey-predator Lotka-Volterra model to explain how the populations of predators and prey fluctuate over time. These equations may be used to estimate how predator and prey populations would fluctuate under various circumstances which mentioned above. By analyzing these models, ecologists may learn about the complex interactions between various species in an ecosystem and create plans for biodiversity preservation and resource management [2]. Therefore, prey-predator models are building blocks of bio and ecosystems. In an ecosystem, millions of species interact with each other. They compete, change, and disperse simply to seek recourses. In their struggle for existence in this universe, they compete with each other and may attain different attributes or shapes of population interactions. Depending upon their specific settings of applications, they can take the forms of resource-consumer, parasite-host, plant-herbivore, etc.
Among prey-predator interactions, predators are those species that kill and eat another species, to be eaten called prey. In this interaction, when the prey population decreases, the predator population also decreases; moreover, when the prey population increases, the predator population also increases. Many models have been proposed to comprehend the process of competition among populations of two species. Among these models, the most well-known model is the following predator-prey model which was proposed by Volterra in 1931 [3]: where the number of prey and predator are denoted by x and y at time t, respectively, whereas all involved model's parameters are positive. This model represents the direct relationship between the prey and predator populations. If predators are absent, then the number of prey grows exponentially; when the prey population decreases, the predator population also decreases exponentially. Moreover, in the model, dxy and bxy denote the predator-prey confrontation, respectively, which are useful to predators and harmful to prey. It is noted here that due to the harvesting effect, model (1.1) becomes [4]: Reaction-diffusion, which is another factor in the interaction of prey-predator, was considered by Lazaar et al.; they studied the local and global dynamics of prey-predator by considering the prey refuge [5]. Saadeh et al. [6] considered the following model to describe different dynamical properties which are effected by commensurate and incommensurate orders: , y n+1 = y n + δy n a − by n x n , (1.3) where a, b, c, and δ are positive parameters. Elettreby et al. [7] studied the dynamics of a discrete prey-predator model with mixed functional response. Many other mathematical models which were considered in continues form showed different results to contribute to the stability of ecosystem. For instance, Chen et al. [8] examined the hopf bifurcation of a species interaction model. Chen and Wu [9] investigated the dynamics of a diffusive predator-prey model with a harvesting policy and a network connection. Chen and Srivastava [10] investigated the bifurcating solution of a predator-prey model. Chen and Wu [11] studied bifurcation in the Previte-Hoffman model. Chen and Wu [12] studied the dynamics of a diffusive predator-prey model. Britton [13] suggested the following Leslie's model: where all involved model's parameters are positive. The British mathematician Patrick Leslie created the Leslie model in the 1940s for examining the dynamics of fish populations. Leslie was fascinated by how populations of fish and other species changed over time, and he realized that a mathematical model may assist him in forecasting these changes. According to the Leslie model, the population may be split into several age groups, and the birth and death rates of people in each group can be used to forecast how the population will change over time. Since then, a variety of creatures have been studied using the model, making it a crucial tool in ecology and conservation biology. The following assumptions constitute the foundation of the Leslie model [14,15]: There are discrete age groups within the population; the birth rate is consistent throughout all age groups; the mortality rate is consistent among all age groups; and there is no inward or outward population migration.

Problem statement
Predator-prey models and other population dynamics are studied using the Leslie model; it considers elements including population interactions, mortality rates, and birth rates. The model is helpful for understanding the effects of environmental variables, researching long-term ecosystem dynamics, and anticipating changes through time. Moreover, it is well known that when the populations have nonoverlapping generations, the discrete models defined by maps are more logical than the continuous models. Additionally, the discrete-time models offer richer dynamics and more capable computational results for numerical simulations than the continuous ones [16][17][18][19]. Therefore, in the present study, we will study the hybrid control and bifurcations of the following discrete Leslie's model: which is a discrete analogue of (1.4) by the Euler forward formula.

Layout of the paper
The layout of rest of the paper is as follows. The stability analysis at fixed points is given in Section 2. In Section 3, bifurcations at equilibria are given, whereas Section 4 describes the control of the N-S bifurcation. In Section 5, the obtained results are verified numerically, whereas a concise summary is provided in Section 6.

Stability analysis at equilibria
In this study, we explore stability analysis at equilibria of the discrete Leslie's model (1.5) by the stability theory [20][21][22][23][24][25]. In this respect, we first find equilibria of the discrete Leslie's model (1.5) by algebraic techniques. By definition of a fixed point, if E xy (x, y) is the equilibrium point of the model (1.5), then at E xy (x, y) under the map is as follows: Now, in subsequent sections, we will explore the stability analysis at E x0 a b , 0 and E + xy aα cd+bα , ad cd+bα .

Stability analysis at
whose characteristic roots are From (2.10), the dynamics of the Leslie's model (1.5) at E x0 a b , 0 can be summarized as

Stability analysis at E
14) whose characteristic equation is Lastly, roots of (2.15) are Hereafter, two Lemmas for ∆ < 0 (∆ ≥ 0 are presented to show the complete topological classifications at E + xy aα cd+bα , ad cd+bα of the discrete model (1.5). .

Bifurcation analysis
The possible bifurcation analysis are explored in this section at fixed points E x0 a b , 0 and E + xy aα cd+bα , ad cd+bα . First, we will study the bifurcation analysis at E x0 Therefore, in the following theorem, we are going to study the detailed indicated bifurcation analysis at E x0 a b , 0 .

Control of N-S bifurcation
In prey-predator Leslie models, controlling the Neimark-Sacker bifurcation is essential since it enables us to stabilize the dynamics of the populations. A sort of bifurcation called a Neimark-Sacker bifurcation occurs when a stable limit cycle, which is a population dynamics repeating pattern, loses stability and gives rise to a new, unstable limit cycle. Chaos in dynamics can result from this, which can be challenging to forecast and manage. In prey-predator Leslie models, we may prevent chaotic dynamics and stabilize the populations by managing the N-S bifurcation. As a result, we are able to better anticipate and lessen the consequences of environmental variables like habitat loss, climate change, and other disturbances. This is crucial for managing natural resources and protecting bio-diversity. In prey-predator Leslie models, the N-S bifurcation needs to be controlled in order to comprehend long-term ecosystem dynamics and create management plans for natural resources. We may learn more about the intricate interactions between many species in an ecosystem, establish plans for biodiversity preservation, and manage resources by researching the interactions between predator and prey populations. Furthermore, by limiting the N-S bifurcation, we can forecast how ecological changes in populations will impact it over time and create plans to reduce the impact of disturbances. Therefore, hereafter we will control the N-S bifurcation by the hybrid control method motivated from existing literatures [32][33][34]. If β is a bifurcation parameter, then the Leslie's model (1.5) becomes x n+1 =β (1 + ah)x n − bhx 2 n − chx n y n + (1 − β)x n , The roots of (4.2) with β are which is depicted in (4.2), has two conjugate roots with modulo 1. Additionally, it is easy to establish that E + xy aα cd+bα , ad cd+bα of the system (4.1) is non-hyperbolic, stable focus, and unstable if following conditions hold, respectively: and Recall that if a is a bifurcation parameter, then the system (4.1) becomes x n+1 = β (1 + (a * + ϵ)h)x n − bhx 2 n − chx n y n + (1 − β)x n , y n+1 = β (1 + dh) x n y n − αhy 2 n x n + (1 − β)y n , (4.8) where characteristics roots are (4.9) Furthermore, for the system (4.1) to undergo N-S bifurcation, the following non-degenerate condition(s) should hold: Furthermore, it is also require that λ υ 1,2 1, υ = 1, · · · , 4 which corresponds to T (β) −2, 0, 1, 2 if ϵ = 0. This compute to get α chd 2 (β 2 hd−2) b(3βhd−2−β 2 h 2 d 2 −hd) , chd 2 (β 2 hd−1) b(2βhd−1−β 2 h 2 d 2 −hd) . Now, by (3.13), the system (4.1) becomes with x * = (a * +ϵ)α cd+bα and y * = (a * +ϵ)d cd+bα . Now, the system (4.11) becomes u n+1 = δ 11 u n + δ 12 v n + δ 13 u 2 n + δ 14 u n v n , v n+1 = δ 21 u n + δ 22 v n + δ 23 u n 2 + δ 24 u n v n + δ 25 v n 2 + δ 26 u n 3 + δ 27 u n 2 v n + δ 28 u n v n 2 , (4.12) where (4.13) Now, using the following translation, u n v n := δ 12 0 η − δ 11 −ξ U n V n , (4.14) system (4.11) gives U n+1 = ηU n − ξV n + F 1 (U n , V n ) , where F 1 (U n , V n ) = β 11 U 2 n + β 12 U n V n ,  and (4.18) From (4.16), one gets:     Figure 6b-6f). Finally, N-S bifurcation diagrams are presented in Figure 7 alongside the associated Maximum Lyapunov exponent, demonstrating the accuracy of the results in Sections 2 and 3. Using (5.6) and (5.4) in (3.23), we get ψ = −0.9756591603967086 < 0. This implies that the controlled model (4.1) undergoes a supercritical hopf bifurcation (see Figure 8). To conclude, N-S bifurcation diagrams are presented in Figure 9 alongside the associated Maximum Lyapunov exponent, demonstrating the accuracy of the results in Section 4.    7c Maximum Lyapunov exponents correspond to 9a,9b.

Conclusions
In the current study, we investigated the hybrid control and bifurcation analysis of a discrete Leslie's model (1.5). More precisely, for all model's parameters, it is shown that Leslie's model (1.5) has boundary and interior equilibria E x0 a b , 0 , E + xy aα cd+bα , ad cd+bα , respectively. Additionally, we investigated the topological classifications at equilibria E x0 a b , 0 and E + xy aα cd+bα , ad cd+bα whose main finding are given in Table 2. Moreover, we studied the existence of possible bifurcations at equilibria . In Leslie models, the development of a Neimark-Sacker bifurcation can result in unexpected and chaotic population dynamics, making it challenging to predict the ecosystem's long-term dynamics. However, by comprehending the fundamental causes of N-S bifurcations, we may create plans for population stabilization and environmental impact reduction. Changes in birth and death rates, interactions between predator and prey populations, and environmental issues like habitat loss and temperature change are some of the underlying reasons that can cause N-S bifurcations in Leslie models. These elements have the potential to disrupt population dynamics, resulting in unpredictably erratic population growth. We can create plans for protecting bio-diversity and managing natural resources by researching these aspects. Furthermore, we have studied bifurcation analysis by bifurcation theory. Additionally, we have controlled the N-S bifurcation by employing a hybrid control method. Finally, numerical simulations are also presented to validate the results. h(bdhα+chd 2 −2bα) ; unstable node if a > (cd+bα)(2hd−4) h(bdhα+chd 2 −2bα) ; non-hyperbolic if a = 2(chd 2 +bhdα)−4(cd+bα) bdh 2 α−ch 2 d 2 −2bhα .

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