Dynamics on Effect of Prey Refuge Proportional to Predator in Discrete-Time Prey-Predator Model

Prey-predator models with refuge effect have great importance in the context of ecology. Constant refuge and refuge proportional to prey are the most popular concepts of refuge in the existing literature. Now, there are new different types of refuge concepts attracting researchers. *is study considers a refuge concept proportional to the predator due to the fear induced by predators. When predators increase, fears also increase and that is why prey refuges also increase. Here, we examine the influence of prey refuge proportional to predator effect in a discrete prey-predator interaction with the Holling type II functional response model. Is this refuge stabilizing or destabilizing the system? *at is the central question of this study. *e existence and stability of fixed points, Period-Doubling Bifurcation, Neimark–Sacker Bifurcation, the influence of prey refuge, and chaos are analyzed.*is work provides the bifurcation diagrams and Lyapunov exponents to analyze the refuge parameter of the model. *e proposed discrete model indicates rich dynamics as the effect of prey refuge through numerical simulations.


Introduction
In the current biomathematical literature, prey-predator interaction has become an exciting subject matter due to its influence on the environment. Many researchers [1][2][3][4] have devoted considerable time to explore several perspectives of the dynamical behaviour of this subject matter in ecology and the associated growth of population models [5][6][7]. Nature offers some amount of protection to some prey populations in the form of refuge dimensions. In some instances, refugia give a prolonged prey-predator interaction [8,9] to reduce the probability of extinction due to predation activity. is phenomenon has been investigated extensively by many researchers in the discrete domain of refuse concepts [10][11][12].
In the context of populations characterized by overlapping generations, the nature of the birth process is a continuous matter; therefore, the predator-prey relationship is frequently developed through different models utilizing a deterministic approach such as ordinary differential equations. Several species, including monocarpic plants and semelparous animals, possess a discrete nonoverlapping generation character and have birth and regular breeding seasons.
eir interactions are presented in the form of difference equations or other forms such as discrete-time mappings [13][14][15][16][17]. A discrete-time prey-predator model is usually characterized by more complicated dynamics behaviour than the associated continuous-time models [18][19][20][21]. Several researchers have focused on this concept to present a comprehensive rich dynamics of this phenomenon, including stability analysis of equilibria [22][23][24][25][26][27], Period-Doubling Bifurcation [28], Neimark-Sacker bifurcation [29], and chaos control [30]. Liu and Cai [31] explored the presence of bifurcation and chaos in a discretetime prey-predator system. Elettreby et al. [32] used mixed functional response to investigate the dynamics of a discrete prey-predator model. AlSharawi et al. [33] studied a discrete prey-predator model with a nonmonotonic functional response and strong Allee effect in prey. Wang and Fečkan [34] studied the dynamics of a discrete nonlinear preypredator model. Rech [35] investigated two discrete-time counterparts of a continuous-time prey-predator model. Khan and Khalique [36] discussed Neimark-Sacker bifurcation and hybrid control in a discrete-time Lotka-Volterra model. For some more dynamical investigations related to different versions of prey-predator models, we refer to Elsadany et al. [37]; Baydemir et al. [38]; Rozikov and Shoyimardonov [39]; Singh and Deolia [40]; and references therein.
In this paper, we consider a refuge term proportional to predator density due to fear induced by predators. e earlier literature has pointed out that the use of refuge by a fixed number or a small percentage of prey exerts a stabilizing effect on the dynamics of the interacting populations.
is article is concerned with the above statement assuming that the quantity of prey in refugia is proportional to that of predators due to the fear induced by the predator. We analyze the dynamic properties of such a prey-predator model with prey self-limitation. e rest of the paper is partitioned as follows. A discrete-time prey-predator model with refuge is formulated in the second section. Section 3 is devoted to the local stability analysis of the model. Bifurcation analysis of the proposed model is presented in Section 4. Section 5 deals with the influence of prey refuge on the proposed model. Section 6 gives a chaos control procedure. Section 7 presents numerical simulations with support of the presented dynamical analysis of the proposed model. Section 8 presents the conclusion that is the last aspect of the findings from this study.

Development of a Discrete Prey-
Predator System e densities of prey and predator populations vary with time, have a uniform distribution over space, and have no stage structure for neither prey nor predators. e proposed model takes into consideration a generalised prey-predator model incorporating the logistic growth of prey with functional response φ(x) of the predator population as (dx/dt) � rx(1 − (x/k)) − cφ(x)y, and the predator as a population is epitomised by the expression (dy/dt) � dφ(x)y − fy, where x and y are the density of prey and predator populations correspondingly at any time t. e parameters r, k, c, d, and f are all taken to be positive constants and have its appropriate biological interpretations, respectively. r denotes the intrinsic per capita growth rate of prey population, k represents the environmental carrying capacity of prey population, c denotes the maximal per capita consumption rate of predators, d considers the efficiency with which predators transform consumed prey into new predators, and f represents the per capita death rate of predators. φ(x) deals with the functional response of the predator population and fulfils the assumption φ(0) � 0 and φ′(x) > 0 for x > 0. ere exists a quantity x r of prey population that incorporates refuges, and therefore we have modified the functional response to incorporate prey refuges, and we consider φ(x) � (x/(e + x)), and x r � by. e mathematical form of the proposed predator-prey model is as follows: To arrive at a discrete-time model from (1), let (dx/dt) � ((x t+h − x t )/h) and (dy/dt) � ((y t+h − y t )/h), where x t and y t are the densities of the prey and predator populations, respectively, in discrete time t. Let h ⟶ 1 and f � 1, we obtain the equations for the (n + 1)th generation of the prey and predator populations substituting t by n as follows: x n − c x n − by n y n e + x n − by n , y n+1 � d x n − by n y n e + x n − by n . (2) Let (r/k(r + 1)) � 1 and (r + 1) � a, we obtain the discrete-time predator-prey system from (2) as follows: x n+1 � ax n 1 − x n − c x n − by n y n e + x n − by n , y n+1 � d x n − by n y n e + x n − by n , where a, b, c, d, and e are all positive constants. By the biological meaning of the model variables, we only consider the system in the region Ω � (x, y): x ≥ 0, y ≥ 0 in the (x, y) − plane.

Fixed Points and Stability Analysis
Fixed points of the discrete system (3) are determined by solving the following nonlinear system of equations: We get three nonnegative fixed points by solving above equations: (i) P 0 � (0, 0), (ii) P 1 � (((a − 1)/a), 0), a > 1, and (iii) P 2 � (x 2 , y 2 ); here, x 2 is positive root of the equation where and e characteristic equation of the matrix J is erefore, the discrete-time system (3) possesses the following: e dynamical behaviour of the system is studied utilizing the Jacobian matrix at the fixed point P 0 (0, 0). e Jacobian

Stability and Dynamic
Behaviour at P 1 . e Jacobian matrix at the fixed point

Bifurcation Analysis
In this section, we investigate the existence of NSB and PDB at the positive fixed point P 2 (x 2 , y 2 ) of the discrete-time of (3). Countless numbers of dynamical properties of the system can be analyzed with regard to the emergence of NSB and PDB. Bifurcation frequently occurs when there is a change in the stability of a fixed point. us, the qualitative properties of a dynamical system vary. e NSB and PDB are explored in the positive fixed point direction P 2 (x 2 , y 2 ) of system (3) considering refuge parameter b as a bifurcation parameter. Utilizing the emergence of NSB, closed invariant circles are obtained. e bifurcation can be classified as supercritical or subcritical, leading to a stable or unstable closed invariant curve, in that order. A PDB in a discrete dynamical system is a bifurcation where a small perturbation in a parameter value brings about the system moving to a new behaviour with the double period of the original system.

Neimark-Sacker Bifurcation.
Here, we examine the NSB of the discrete prey-predator model (3) at P 2 (x 2 , y 2 ) for the parameters lying in the following set

Complexity
In exploring the NSB, b is utilised as the bifurcation parameter. Further, b * is the perturbation of b where |b * | ⋘ 1, and perturbation of the model is considered as follows: Let u n � x n − x 2 and v n � y n − y 2 , then equilibrium P 2 (x 2 , y 2 ) is converted into the origin, and further making expansion f and g as a Taylor series at (u n , v n ) � (0, 0) to the third order, model (7) turns to where Note that the characteristic equation associated with the linearization of model (8) at (u n , v n ) � (0, 0) is given by To study the normal form, let c � Im(λ 1,2 ) and δ � Re(λ 1,2 ). We define T � 0 1 c δ , and using the trans- x n+1 � δx n − cy n + f 1 x n , y n , y n+1 � cx n + δy n + g 1 x n , y n , where the functions f 1 and g 1 denote the terms in model (9) in variables (x n , y n ) with the order at least two.
To undergo NSB, we require that the following discriminatory quantity Ω be nonzero:

Complexity
Finally, the above analysis leads to the following associated results.
, and it is obvious that T is invertible. Using the transformation u n v n � T x n y n , system (13) becomes where the functions f 1 and g 1 denote the terms in model (14) in variables (u n , v n , b * ) with order at least two. Using the centre manifold theorem, we know that there exists a centre manifold W c (0, 0, 0) of system (14) at (0, 0) in a small neighbourhood of b * � 0, which can be approximately described as follows: where Complexity 5 System (14) which is restricted to the centre manifold W c (0, 0, 0) has the following form: where For PDB, the two discriminatory quantities ξ 1 and ξ 2 be nonzero, Finally, from the above analysis, we have the following result.

Influence of Refuge
is section presents the impact of prey refuge on the model when the coexistence steady-state point P 2 (x 2 , y 2 ) exists and is stable. It is simple to observe that x and y are continuous differential functions of parameter b.

Complexity
Clearly, if 0 < (dx 2 /db) < y 2 , then x 2 is strictly increasing function of the parameter b and y 2 is strictly decreasing function of the parameter b.
If (dx 2 /db) < 0, then x 2 and y 2 are strictly decreasing function of the parameter b.

Chaos Control
In discrete-time models, several approaches can be used in chaos control. ese include the following: hybrid control method, pole-placement technique, and the state feedback method [9,14,18,30]. In this regard, a feedback control method is employed to stabilize the chaotic orbits at an unstable positive fixed point of model (3) as follows: x n+1 � ax n 1 − x n − c x n − by n y n e + x n − by n + S, y n+1 � d x n − by n y n e + x n − by n .
With the feedback control law as the control force, S � − q 1 (x n − x 2 ) − q 2 (y n − y 2 ), where q 1 and q 2 are the feedback gain and (x 2 , y 2 ) is a positive fixed point of the model. e Jacobian matrix J for system (20) at (x 2 , y 2 ) is . e characteristic equation of the matrix J is given by Let λ 1 and λ 2 be the eigenvalues, then e lines of marginal stability are determined by solving the equation λ 1 � ± 1 and λ 1 λ 2 � 1. ese conditions guarantee that the eigenvalues λ 1 and λ 2 have modulus less than 1.

Complexity
With the comparison of bifurcation analysis through the pictorial representation in Figure 3 with Figure 1 for the same parameter value with only the difference in functional response parameter e, we observed that Neimark-Sacker bifurcation vanishes from Figure 3, but Period-Doubling Bifurcation occurs. Figure 4 is the corresponding Lyapunov exponent diagram of Figure 3. Furthermore, the bifurcation diagram by changing the parameter value a is shown in Figure 5, and compared with Figures 1 and 3, here both bifurcations are vanished. 10 Complexity x n+1 � ax n 1 − x n − c x n − by n y n e + x n − by n + S, y n+1 � d x n − by n y n e + x n − by n . (27) e Jacobian matrix J for system (27) e roots of (28) are given by λ 1 � − 0.7023 and λ 2 � 0.1081. Now, line l 1 is 0.5245q 1 − 0.4987q 2 � − 1.5874, line l 2 is 0.4755q 1 + 0.4987q 2 � − 1.5068, and line l 3 is 1.5245q 1 − 0.4987q 2 � − 0.6816. e stable eigenvalues lie within a triangular region by the lines l 1 , l 2 , and l 3 shown in Figure 7. e system is stable for the triangular region ABC bounded by the marginal lines l 1 , l 2 , and l 3 .

Conclusion
In this paper, the impacts of refuges used by prey on a preypredator interaction are studied through the analytical approach and graphical representation. is study considered the prey refuge proportional to predator density due to fear induced by a predator on prey. e detailed effects of the local stability of the interior steady-state point and the longterm dynamics of the interacting populations on the model have been investigated. e results show that the impact of refuges can stabilize and destabilize the interior equilibrium point of the proposed prey-predator model for different parameter sets.
Different observations have been made on the types of refuge and their impacts on the proposed system. Generally, in the existing literature, we always found the stabilizing effect of refuge. However, here, we observed an unstable to stable to unstable behaviour in Figure 1. For a different parameter set in Figure 3, we have also found a stable to unstable behaviour of the system. In Figure 5, another interesting observation is that refuge is profitable for prey and adverse effect on the predator. We have studied the bifurcations of the proposed discrete preypredator model with refuge. It is shown that the model exhibits various bifurcations of codimension one, including PDB and NSB, as the values of the parameters vary.

Data Availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.