Dynamics of a Predator-Prey Model with Fear Effect and Time Delay

In this paper, we propose a time-delayed predator-prey model with Holling-type II functional response, which incorporates the gestation period and the cost of fear into prey reproduction. )e dynamical behavior of this system is both analytically and numerically investigated from the viewpoint of stability, permanence, and bifurcation. We found that there are stability switches, and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. )e explicit formulae which determine the direction, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. We perform extensive numerical simulations to explore the impact of some important parameters on the dynamics of the system. Numerical simulations show that high levels of fear have a stabilizing effect while relatively low levels of fear have a destabilizing effect on the predator-prey interactions which lead to limit-cycle oscillations. We also found that the model with or without a delay-dependent factor can have a significantly different dynamics. )us, ignoring the delay or not including the delay-dependent factor might result in inaccurate modelling predictions.


Introduction
Predator-prey interaction is a central topic in ecology and evolutionary biology which has been extensively studied from different aspects by many researchers over the last few decades. Mathematical models have played a vital role in understanding these complex scenarios. Most of the existing predator-prey models are based upon classical Lotka-Volterra formalism, and it is usually assumed that the predators can impact the prey population only through direct killing as direct predation is relatively easy to observe in nature. However, the presence of predator may significantly change the behaviour and physiology of prey to such an extent that it could affect the prey population more effective than direct predation [1][2][3].
In the presence of predators, many prey species change their behaviour due to predation risk and show a variety of antipredator responses, which incorporates foraging activity [4], habitat alterations, vigilance [5] and some physiological changes, etc. [3,6]. Many researchers observed that due to fear of predation risk, the reproduction of the scared prey decreases. For example, in the Greater Yellowstone Ecosystem, wolves influence the reproductive physiology of Elks [7]. Snowshoe hares shift to relatively safe but less profitable microhabitats [8]. Birds react to the sound of predator species with antipredator defenses, and they flee from their nests at the first sight of danger [1]. Other pieces of evidence suggest that fear can also affect populations like mule deer [9] and dugongs [10].
In 2011, Zanette et al. conducted a manipulation on song sparrows during a whole breeding season and showed that there is a 40% reduction in the offspring production of the song sparrows due to the fear from predator [11]. Very few researchers have investigated the impact of fear effect in a predator-prey model with the help of mathematical modelling. Wang et al. first proposed a predator-prey model by incorporating the cost of fear into prey production. eir analysis showed that fear effect can stabilize the oscillation of the system [12]. Wang and Zou formulated a predator-prey model with the cost of fear and adaptive avoidance of predators. e prey population is divided into juvenile and mature stages; this model is represented by a system of delay differential equations with maturation delay [13]. ey theoretically analyzed the model dynamics for constant defense level and adaptive defense level. Numerical simulations showed that both strong adaptation of adult prey and the large cost of fear have destabilizing effect, but large population of predators has a stabilizing effect on the system. Sasmal and Takeuchi studied a predator-prey system with fear effect, group defense, and Holling-type IV functional response; they observed that the model can exhibit multistability [14]. Panday et al. studied the impact of fear in a tritrophic food chain model with Holling-type II functional response and observed that fear can stabilize the system from chaos to stable focus through the period-halving phenomenon, and chaotic dynamics can be controlled by the fear factors [15]. Sarkar and Khajanchi [16] proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling-type II functional response.
eir mathematical analysis suggests that strong antipredator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors.
Inspired by the aforementioned works, the aim of the current work is to develop a predator-prey model, which incorporates the fear effects of predators on prey as well as the time delay. Further, we investigate the impacts of important parameters on the dynamics of system. e rest of the paper is organized as follows. In the next section, we will give the underlying assumptions and formulate the model. In the following section, we present some results on the stability of equilibria and existence of Hopf bifurcation. In Section 4, we investigate the properties of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. We show the permanence of the system in Section 5. In Section 6, we provide some numerical simulation results which reveal some potential roles that the fear effect and the time delay may play in predator-prey interactions. A brief discussion completes the paper.

Model Formulation
Let x(t) and y(t) be the densities of prey and predator populations at time t, respectively. We assume that the prey population follows a logistic growth without any predator population, which can be split into three different parts, namely, the birth rate, the natural death rate, and the density dependent death due to intraspecific competition among preys. Let r be the birth rate of prey, d 1 be the natural death rate of prey, and d 2 be the death rate due to intraprey competition. Since the fear effect will reduce the production, we will multiply the production term of prey rx by a factor f(k, y) � (1/(1 + ky)) which accounts for the cost of antipredator defense due to predation fear. Also, it is reasonable to assume that the production of predators after predation is not instantaneous; for the sake of simplicity, we suppose that the time lag required for gestation of predator is τ.
erefore, our predator-prey model with fear effect is given by Here, the parameter k refers to the level of fear, which is due to the antipredator response of prey. In this study, we adopt the widely used Holling-type II functional response of predators [17], where (m/a) is the predation rate and (1/a) is the half-saturation constant. c is the conversion efficiency from prey biomass to predator biomass and d is the death rate of predator. Since predator goes through a gestation stage, and it endures the death rate, then a factor e − dτ appears in the first term of the second equation of system (1). e parameters r, d 1 , d 2 , m, a, c, and d are all positive constants, and k and τ are nonnegative constants. e initial conditions take the form of where C([− τ, 0], R 2 + ) denotes the space of continuous functions mapping the interval [− τ, 0] into R 2 + and denotes the norm of an element ϕ � (ϕ 1 , It is straightforward to show (see [18]) the following.

Lemma 1. e solution of system (1) with an initial condition (2) is unique and nonnegative for all t ≥ 0.
Note that from the first equation in (1), we obtain (dx/dt) ≤ (r − d 1 )x; by a comparison argument, we see that if r < d 1 , then lim t⟶∞ x(t) � 0, and applying the theory of asymptotically autonomous systems to the second equation in (1), we also have lim t⟶∞ y(t) � 0. at is, when r < d 1 , both prey and predator species go to extinction. So, throughout this paper, we assume that r > d 1 .

Lemma 2.
ere exists an M > 0 such that for any solution of system (1) with an initial condition (2), there must be a T > 0 such that Proof. We first consider the system whose solution satisfies Since (dx/dt) ≤ (r − d 1 )x − d 2 x 2 , then by the comparison principle, we have Hence, for any ε > 0 small enough, there exists a t * > 0 sufficiently large such that 2 Complexity For L(t) � ce − dτ x(t − τ) + y(t) and t > t * + τ, we have It follows that is completes the proof. □ □ Lemma 3 (see [19], Lemma 2.1). Consider the following equation: It is easy to check that (1) has a trivial extinction equilibrium E 0 � (0, 0). is system also has a semitrivial equilibrium E 1 � (((r − d 1 )/d 2 ), 0) under the condition r > d 1 . To find the positive (coexistence) equilibrium E * � (x * , y * ), letting the right-hand side be zero, we then get x * � (d/(mce − dτ − da)), which exists if mce − dτ > da, and y * is a positive root of the equation . Clearly, (10) has one positive root y * if and only if d 1 + d 2 x * − r < 0; this inequality is equivalent to en, we conclude that (1) has a unique positive equilibrium E * � (x * , y * ) if (11) holds, and the expression for y * is as follows:
Proof. For equilibrium E 0 � (0, 0), the characteristic equation is Obviously, (13) has one positive root r − d 1 and one negative root − m. Hence, E 0 is a saddle point. e characteristic equation at E 1 � (((r − d 1 )/d 2 ), 0) is as follows: Let )) e − λτ ; then it is clear that (14) has one root d 1 − r < 0 and the other roots are determined by the equation f(λ) � 0.
Next, we will show that E 1 is globally attractive if (11) is reversed. Since mce − dτ < da + (dd 2 /(r − d 1 )), we can choose small enough ε > 0 such that From (6), we know that for t > t * . But for the equation by (17) and Lemma 3, it follows that lim t⟶∞ V(t) � 0. erefore, an application of the standard comparison argument yields lim t⟶∞ y(t) � 0. en, the limiting equation , and E 1 is globally asymptotically stable in C.
Note also that if (11) holds, then Hence, f(λ) � 0 has at least one positive root and E 1 is unstable.
□ eorem 1 implies that E 1 loses its stability to the occurrence of the positive equilibrium E * when (11) holds. e local stability of E * is addressed in the following theorem.
Let λ � iω, ω > 0 is a root of (21); then, (21) implies Squaring and adding both equations of (24), it follows that where z � ω 2 . A straightforward computation shows that (26) Now, we can summarize the above result on the local stability of the positive equilibrium E * . Theorem 2. Assume that (11) (21) has no purely imaginary root, and E * is locally asymptotically stable for all τ ≥ 0. (ii) If m 4 > m 2 > 0, then (21) has a unique pair of purely imaginary roots ± iω 0 , and E * may lose stability as τ increases from zero, (25).
To further investigate the stability of E * under condition (ii) of eorem 2, we need to check the transversality condition at λ � ± iω 0 . Since x * and y * are both time delaydependent, it is hard to discuss the transversality condition, if we suppose that the coefficient in conversing prey into predator ce − dτ � c 0 is a constant, independent of delay τ, then we can extend the result in eorem 2 as follows.
Proof. From (24), we know that τ n corresponding to ω 0 is

Complexity
To complete the proof of eorem 3, we only need to show that Differentiating (21) with respect to τ, we have and hence, en, we have (d(Reλ)/dτ)| τ�τ n > 0. erefore, the transversality condition holds, and the Hopf bifurcation occurs at τ � τ n .

Properties of Hopf Bifurcations
In this section, we will study the properties of the Hopf bifurcations by using the normal theory and the center manifold theorem due to Hassard et al. [20]. Let and dropping the bars for simplification of notations, system (1) is transformed into a functional differential equation in where and where By the Riesz representation theorem, there exists a function η( θ, v) of bounded variation for θ ∈ [− 1, 0], such that In fact, we can choose where δ denotes the Dirac delta function. For where and a bilinear inner product where η(θ) � η(θ, 0). en, A(0) and A * are adjoint operators. By the discussion in Section 3, we know that ±iω 0 τ n are eigenvalues of A(0). Hence, they are also eigenvalues of A * . We first need to compute the eigenvectors of A(0) and A * corresponding to iω 0 τ n and − iω 0 τ n , respectively. Suppose q(θ) � (1, α) T e iω 0 τ n θ is the eigenvector of A(0) corresponding to iω 0 τ n ; then A(0)q(θ) � iω 0 τ n q(θ). en, from the definition of A(0), (34), (37), and (38), we have Similarly, we can obtain the eigenvector q * (s) � D(1, α * )e iω 0 τ n s of A * corresponding to − iω 0 τ n , where

Permanence
In order to prove the uniform persistence of system (1), we need the uniform persistence theorem for infinite dimensional systems from [24]. Let X be a complete metric space.
Suppose that X 0 is open and dense in X, with X 0 ∪ X 0 � X,

Let T b (t) � T(t)| X 0 and let A b be the global attractor for
en, X 0 is a uniform repeller with respect to X 0 , i.e., there is an ϵ > 0 such that for any Now, we consider the persistence of system (1).

Theorem 4. Let (11) hold; then there exists an ε > 0 such that every solution x(t), y(t) of system (1) with initial conditions
Proof. Let It suffices to show that there exists an ε > 0 such that for any solution u t of system (1) initiating from X 0 ,lim inf t⟶∞ d(u t , X 0 ) ≥ ε. To this end, we verify below that the conditions of the above uniform persistence theorem are satisfied. Firstly, we show X 0 is positively invariant. By (1), we have and en, x(t) > 0 and y(t) > 0 for all is implies that X 0 is positively invariant.
Similarly, we can show that X 0 is positively invariant, and condition (48) is satisfied. We have verified the point dissipativeness of the semiflow of system (1) in 2.
Denote the ω− limit set of the solution of system (1) For any given (ϕ 1 , Case 2. y(0) > 0. en we must have x(0) � 0; from the first equation of (1), we have x(t) ≡ 0 for all t ≥ 0; again from the second equation of (1), we have y(t) ⟶ 0 as t ⟶ ∞.
Finally, we see that for any given us, Ω � E 0 , E 1 . If invariant sets E 0 and E 1 are isolated, then Ω is isolated and is an acyclic covering. e isolated invariance of E 0 and E 1 will follow from the following proof.
We now show that where W s (E 0 )(W s (E 1 )) denotes the stable manifold of E 0 (E 1 ). Suppose W s (E 0 ) ∩ X 0 ≠ ∅; then there exists a solution (x(t), y(t)) ∈ X 0 ,t ≥ 0, of (1) with initial conditions in X 0 , such that Since r > d 1 , we can choose sufficiently small η 1 > 0 so that Lett 0 > 0 sufficiently large such that 8 Complexity (57) en, for t > t 0 , we have Consider the following equation: By the comparison principle, we have x(t) ≥ W(t) for all t ≥ t 0 . It is easy to see that (59) has a positive equilibrium W * � (((r/1 + kη 1 ) − d 1 − mη 1 )/d 2 ), which is globally attractive. Note that x(t) ≥ W(t) for all t ≥ t 0 and lim t⟶∞ W(t) � W * ; this is a contradiction to (55).
is completes the proof.
In Lemma 2, we have shown that the solutions of system (1) are ultimately bounded; together with eorem 4, we now give the main result in this section.

Numerical Simulations
In order to investigate the role of time delay and fear in our model (1), we perform some numerical simulations to illustrate the analytical results observed in the previous sections. All the simulations are carried out in MATLAB.
First, we choose the parameter values as where most of the parameters are taken from [12]. For the above set of parameter values, in Figure 1, we observe that the system shows a limit cycle around the interior equilibrium E * . If we further increase the value of τ, for τ � 0.4, we see that the solutions approach the coexistence equilibrium; see Figure 2.
To observe the long-term behavior for a range of values of τ, in Figure 3, we draw the bifurcation diagram with respect to τ. From Figure 3, we can see that as we increase the value of τ, the prey and predator populations show oscillatory coexistence for the small values of τ; then, both populations show stable coexistence.
is indicates that increasing delay τ may induce a transition from the state where the populations of the prey and predator oscillate Complexity 9 periodically to a stability situation. And also, we note that the prey population increases, while the predator population decreases as we gradually increase τ, and finally the predator population goes to extinction; the system becomes stable around the semitrivial equilibrium E 1 .
To investigate the impact of fear when the system is unstable around the positive equilibrium, we fix τ � 0.2. We draw the bifurcation diagram considering k as a bifurcation parameter. Figure 4 shows that as we increase the level of the fear effect in prey, the coexistence of prey-predator changes from stable periodic oscillations to stable equilibrium. Figure 4 indicates that the prey population tends to a steady state. e fear effect will not affect the prey population over the long term. Biologically, after a certain level of fear with prey population, the fear has no effect due to physiological impact when the prey populations are habituated with fear from predator species, while the predator population eventually will decrease with increasing k.
If we choose parameter values as those in Figure 1 and let m vary in the interval [0.3, 0.65], Figure 5 gives the If the value of the predation rate is low, then the predator population cannot survive. As we increase the value of m, the coexistence of prey-predator changes at stable equilibrium to stable oscillatory coexistence.
To investigate the properties of the bifurcating periodic solution, we neglect the delay-dependent factor e − dτ ; that is, we consider the following system: We use the following parameter values:  stable when τ � 0. Take n � 0; we then have ω 0 � 0.2504, τ 0 � 0.7003, μ 2 � 0.1746 > 0, β 2 � − 0.0081 < 0, and T 2 � 0.2383 > 0. us, E * is stable for τ < τ 0 as illustrated in Figure 6. When τ passes through the critical value τ 0 , E * will lose its stability and a Hopf bifurcation will occurs; that is, a family of periodic solutions bifurcate from E * . Since μ 2 > 0, β 2 < 0, then the Hopf bifurcation is supercritical and its direction is τ > τ 0 . ese bifurcating periodic solutions from E * at τ 0 are stable; see Figure 7.
In Figure 8, we explore the effects of delay τ on the dynamics of system (64). Figure 8 shows that as τ increases, at first the positive equilibrium is stable; then when τ is greater than a certain critical value, periodic solution occurs; this is coincident with eorem 3, which implies that delay destabilizes the equilibrium and induces oscillatory dynamics. By comparing Figure 3 and Figure 8, we see that the difference is big in the dynamics when the delay-dependent factor is incorporated or not.
In Figure 9, taking k as a bifurcation parameter, we found that the fear effects can stabilize the system. In Figure 10, we draw the bifurcation diagram with respect to m; the shapes of the bifurcation diagrams for system (64) in Figure 10 and the original system (1) in Figure 5 are very similar.

Discussion
In this paper, taking into account the time delay corresponding to the gestation period and the effect that the fear of predators has on prey, we have proposed a predator-prey model with Holling-type II functional response. We have shown that the boundary equilibrium E 1 is globally asymptotically stable if (11) is reversed and pointed out that the instability of E 1 implies the permanence of the system   and existence of the positive equilibrium E * . Furthermore, for the system without delay-dependent factor, by choosing the delay τ as a bifurcation parameter, sufficient conditions for the existence of Hopf bifurcation have been presented. By using the theory of normal form and center manifold theorem, the explicit formulae determining the direction, stability, and other properties of bifurcating periodic solutions have been derived. e novelty of this paper is that we have incorporated both the fear effect and a delay-dependent factor that ensures the predators that do not survive the delay interval do not contribute to the growth of the predator population. Our investigation shows that the dynamics of system (1) are affected significantly by not only the biomass transfer delay, but also the level of the fear effect. In [25], the delay-dependent factor was neglected.
Numerical simulations are performed to show the roles that time delay and fear effects can play in predator-prey interactions. For the original system (1), we have shown that both large time delay and high levels of fear (or equivalently strong antipredator responses) can stabilize the predatorprey system by excluding the existence of periodic solutions. And also, we observe that if the time delay as well as the fear level increases, the predator population goes to extinction. While for system (64) which neglects the delay-dependent factor, numerical simulations show that the delay has destabilizing effect while the large cost of fear has a stabilizing effect on the predator-prey interactions.   indicates that ignoring the delay-dependent factor results in model in (63) that does not predict the extinction of the predators, even if the delay is very large. erefore, time delay and fear effect significantly influence the dynamics of the system and play important roles in ecosystems stability.
In the current paper, we assumed that the fear effects only reduce the birth rate of the prey, but it may also increase the death rate of adult prey due to long-term physiological impacts [11]. To make the model more realistic, some other important mechanisms can also be incorporated; for example, spatial effect is an important factor due to foraging behaviors of both prey and predator.
e functional response plays a vital role in determining the different complex dynamical behaviors; different functional response functions can be chosen to reflect the group defense. It would be interesting and challenging to consider the aforementioned possible extensions; we leave these problems for further investigation.

Data Availability
All data generated or analyzed during this study are included in this article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.