Predator-Prey Model with Refuge, Fear and Z-Control

: In this paper, we consider a predator-prey model incorporating fear and refuge. Our results show that the predator-free equilibrium is globally asymptotically stable if the ratio between the death rate of predators and the conversion rate of prey into predator is greater than the value of prey in refuge at equilibrium. We also show that the co-existence equilibrium points are locally asymptotically stable if the value of the prey outside refuge is greater than half of the carrying capacity. Numerical simulations show that when the intensity of fear increases, the fraction of the prey inside refuge increases; however, it has no effect on the fraction of the prey outside refuge, in the long run. It is shown that the intensity of fear harms predator population size. Numerical simulations show that the application of Z-control will force the system to reach any desired state within a limited time, whether the desired state is a constant state or a periodic state. Our results show that when the refuge size is taken to be a non-constant function of the prey outside refuge, the systems change their dynamics. Namely, when it is a linear function or an exponential function, the system always reaches the predator-free equilibrium. However, when it is taken as a logistic equation, the system reaches the co-existence equilibrium after long term oscillations.


Introduction
redator-induced stress in prey animals has not been much studied by physiologists or population ecologists. Canon [1] studied the concept of predator induced stress in a prey population. Population ecologists have focused their studies on predator induced stress effects on the birth rate of free-living prey populations and found that it affects the demography process of prey animals seriously. By experiments in laboratories and field studies it has been demonstrated that mere exposure of prey animals to predators affects the birth rate of prey, and this phenomenon of behavioral change of demography is called the "Ecology of Fear", "Degree of Fear" or "Cost of Fear" by ecologists. Only a little work has been done on the subject of stress induced in a prey population due to predators. The concept of stress was limited to humans and it was thought that stress in a prey population is transitory and that it is not lifelong. Ecologists in the 1990's [2][3][4] verified experimentally that fear of predators is a more powerful cause of demographical change in a prey population than direct killing, shortage of food, or parasitic infection. Fear of predators in prey persists even in the absence of predators and has long-lasting effects on their production of prey species. Sapolsky [5] explained the concept of stress in zebras due to fear of lions. Zanetteet al. [6] reported, after experimental verification, that the sparrow reduces offspring production by 40% just with intimidation by predators where direct killing is stopped by some means. Zanetteet al. [6], Eggers et al. [7], and Travers et al. [8] verified that female sparrows lay fewer eggs and, due to incubation disruption, fail to hatch eggs. Due to fear, they bring less food to their nests and, as a result, a greater proportion of their nestlings starve to death. Creel et al. [9], Creeland Christianson [10], and Creel et al. [11] reported that in the National Parks, USA, due to intimidation by wolves, elk pregnancy rates decline. Recently, Moncluset al. [12] examined an association between predator risk and birth rate of prey.
Field studies have demonstrated that playback of predator sounds can affect the emotions of prey. Remage-Healey et al. [13] demonstrated that a playback sound of dolphins affects the emotions of gulf toad fish. Mateo [14] found that playback of the call of predators' alerts squirrels and that they communicate predator risk to each other. The playback sound of predators increases the glucocorticoid level in prey and hence increases the fear or stress in the prey population. Wanget al. [15] studied how fear of a predator reduces the reproduction of prey animals and found that it could destabilize the system.
A refuge is an area, such as island, where wild animals obtain protection from predation. In this protected territory the chances of being hunted by predator are reduced and these areas reduce the chances of extinction of prey species due to predatory killing. It is a natural phenomenon of prey species in an ecosystem to seek protection from predation (Cowlishaw [16], Sih [17]). Refuge habitats are of different types, such as burrows, trees, cliff faces, or dense vegetation (Clarke et al. [18], Dill and Houtman [19], Berger [20], Cassine [21]). Coral reefs provide refuge for prey fish (Friedlander and Martine [22], Sandinet al. [23]).
Various control systems are used to prevent a species from drastic oscillations and avoid extinctions in an ecological system. The literature records various techniques for control in the multi-species Lotka-Volterra System. These controls are called adaptive control, back idea of control, impulsive control and applications of control theory to Lyapunov functions. One can refer to [24,25,26], where Z-control obtains the desired steady state quickly and prevents high amplitude oscillations. The Z-type control method is an error-based dynamic method, and in this method, it is certain that error function converges to zero. The error between desired outputs and actual outputs go to zero exponentially. There are two ways to apply Z-control in a predator-prey system. The first method is called direct control, where both prey and predator are controlled simultaneously to bring the population to a desired level. The second method is called indirect control, where either prey species or predator species is controlled through immigration, emigration or culling. The second species automatically comes to the desired level exponentially.
Our model is similar to t ha t of Wang et al. [27], and we examine this model by introducing predator fear. Prey only come out of refuge when they feel t h e r e i s less predation; otherwise, they go back to the protected area. We use two control measures to bring the model population of prey and predator to a desired level, and thus we can save it from becoming extinct.

Model formulation
We studied a model where a prey species lives in two different habitats. One is called t h e refuge habitat where the prey species is saved from predation. It is assumed that all resources required for the growth of the prey species are available inside the refuge habitat and that their population grows logistically. It is also assumed that t h e prey species is fully protected from predation inside t h e refuge habitat. When pressure of predation fear is released, then the prey species moves to a second habitat outside the refuge and in this habitat t h e prey species can be killed by predators under the law of mass action. As predation fear increases in the prey species due to the presence of predators, the prey species migrates to the refuge habitat. In the absence of a prey species, predators die exponentially, because predators can only consume prey outside the refuge habitat. This predatorprey interaction is modelled by the following diagram and system of differential equations:

Positivity of solutions
Model (1) describes the dynamics of animal populations and therefore it is very important to prove that all quantities will remain positive for all time. We want to prove that all solutions of the model with positive initial data will remain positive for all time t>0. We can easily verify that Hence, all solutions will remain positive for all time.
Proof. Let = 1 + 2 + . Take the time derivative along the solution of model (1) For any positive constant we have: Let ( + 2 ) = .Therefore, we have Independently of the initial conditions. This completes the proof.

Corollary 1:
If > + > 0, then the region is an invariant region for model 1.
Proof. This is a direct conclusion of Proposition 1
(ii) 1 ̅̅̅ = ( , , 0).The prey species survive inside and outside the refuge habitat and the predator goes to extinction.
(iii) 2 ̅̅̅ = ( 1 ̅̅̅ , 2 ̅̅̅ , ̅). All populations survive. Note that at this equilibrium, and using the third equation of system (1), we get: From the second equation of system (1) at equilibrium we have Using the first equation of system (1) at equilibrium and substituting (4) we get , then there will be only one positive root of ̅ of (5), because (0) = ( 2 ̅̅̅̅ − 1) < 0 , and lim ̅→∞ ( ̅) = ∞ , and then by the continuity of ( ̅) and zero-point theorem, ( ̅) = 0 has one positive solution, so there will be a unique positive coexistence equilibrium.

Stability analysis
In this subsection, we examine the stability of the system about the equilibrium points found in the previous subsection.
(a) Stability analysis of the equilibrium (i): Consider a small perturbation about the equilibrium 1 = 1 ̅̅̅ + , 2 = 2 ̅̅̅ + and = ̅ + .Substitutingthese into the system (1), and neglecting products of small quantities, we obtain the stability matrix: The corresponding characteristic equation is: One of the values of is positive, so (0,0,0) is unstable and hence the both populations will never be extinct.
(b) Stability analysis of the equilibrium (ii): The predator free equilibrium 1 ̅̅̅̅ = ( , , 0)of system (1)  Proof. Using the above mentioned, we obtain the stability matrix and the corresponding characteristic equation is All roots will be negative if < , and hence this equilibrium will be stable.
If ( ) is a continuous and bounded function, then we define: (1) with initial conditions 1 = 1 ( ), 2 = 2 ( )and = ( ) , we have Using fluctuation lemma [28], we can say that there is a sequence { }, and when → ∞ we have Adding the first and second equations of system (1), we obtain Taking the limit on both sides Therefore either 1 ∞ = 0or 0 < 1 ∞ ≤ . According to the limit Theorem [29],we get lim →∞ 1 ( ) = . The This inequality is not true also because for stability of 1 ̅̅̅ , we need > . Therefore, the predator free equilibrium is globally asymptomatically stable if > ; i.e. maximum prey population inside the refuge is greater than .
(a) Stability analysis of the equilibrium(iii): Theorem 2. The equilibrium point 2 ̅̅̅ is locally asymptotically stable if 1 ̅̅̅ > 2 . Proof. The stability matrix of the system (1) around the equilibrium point 2 ̅̅̅ is The corresponding characteristic equation is The equation (8) can be written as

Z-control
To achieve predator population and prey population inside and outside the refuge to a desire level, direct Zcontrol strategy is used. The direct Z-control are functions that a r e incorporated in each equation of the system (1). This system is then described as follows Then we define the error functions as 1 − 1 = 1 = 1 , 2 − 2 = 2 = 2 , − = 3 = 3 , where 1 , 2 and are desiredstates of prey inside the refuge, prey outside the refuge and the predator population respectively. These functions decay exponentially with time, i.e. 1 , 2 and 3 tends to zero. For achieving our purpose we adopt ̇1 = − 1 1 ,̇2 = − 2 2 and ̇3 = − 3 3 , with 1 , 2 , 3 > 0.Now we have

Adaptive non-linear control
We start by first non-dimensionalizing the system (1) by using the following transformations: 1 = 1 , 2 = We use non-linear feedback control for the system (14). This system can be represented as Where 1 , 2 and 3 are adaptive nonlinear feedback control functions which will be the functions of 1 , 2 and y. If these feedback functions stabilize the system, then in a n infinitely long time state the variables converge to zero. Let  Using dynamics of unknown estimators (18) in (17) we will find Clearly the system will be globally stable, because ̇< 0.

Numerical simulation
In this section, we performed several numerical simulations for the system (1) to confirm our theoretical results and to acquire more knowledge about its dynamics and general behavior. The parameter values used are listed in the following table, some of them might be changed in order to study their effect: Table 1. Parameter values used for simulations.

Effect of adaptive control
To study the effect of adaptive control on the system, we look at the stable co-existence equilibrium point. (Figure 2) shows that without the adaptive control the system took a long time to converge to this stable equilibrium point. However, with the use of adaptive control it is clear that the time to reach the equilibrium point is very short, and the system almost instantly started to reach this stable equilibrium point, as seen from (Figure 3).  To study the effect of the intensity of fear, we simulate our model with parameter values taken from Table1, and the intensity of fear taken between 0.1 to 100. It is clear that when the intensity of fear increases, the fraction of the prey in the refuge increases. However, it has no impact on the fraction of prey outside the refuge, as shown from Figures 4-5. On the other hand, when the intensity of fear increases, the fraction of predators decreases as shown in ( Figure6), which dictates that fear is not in the favor of the predator.  show that with the help of z-type control all three populations, t he prey in the refuge, the prey out of the refuge and the predators reaches the desired states as indicated. Clearly the time needed to reach the desired states is very short, and this is due to the power of z-type control, which takes the output to the desired state rapidly.            show that Z-type control could be used to achieve a periodic desired state. From a biological point of view, it is very important to be able to reach a stable limit cycle instead of a constant equilibrium point, as periodic solutions (i.e. limit cycles) are of great interest for ecosystems and more generally for conservation biology. For the purpose of simulation, we consider the desired state to be of the form 1 = 1 + 1 cos 100 2 = 2 + 2 cos 200 = 3 + 3 cos 100

Effect of different forms of
It is more realistic than not to assume the refuge size (i.e. ) is not constant. Figures 21-23 show the fractions of prey in the refuge, the prey outside the refuge and the predator, where we took different forms of . It is clear that when is taken as a linear function (i.e. ( ) = ( + 0 ) 2 ) or as an exponential function of the prey outside the refuge (i.e. ( ) = 2 exp( 0 ) ), then both the prey outside the refuge and in the refuge reach their stable equilibrium and the predator goes to extinction. However, when is taken as a logistic function of the prey outside the refuge (i.e. ( ) = 0 2 (1 − 2 )), then all three populations co-exist together after initial small oscillations. Note that 0 is a positive constant and represents the base-line value for .

Conclusion
All animals are threatened by predators and face the risk of predation. Prey populations change their behavior due to their fear of predators. In this paper we have studied the dynamics of predator-prey interaction where prey reside in two habitats, namely refuge and out of refuge. In refuge, the prey is safe from predatory killing and has sufficient resources to survive, and the population in the refuge grows logistically. Out of refuge, predators interact with the prey and may kill them. Prey live under the fear of predation, but when predator fear is diluted, prey come out of their refuge. On increasing predation fear, prey take shelter in the refuge.
We obtain three biologically feasible equilibria and discuss their stability. The equilibrium free from prey and predator population will always exist and it is unstable i.e. prey, and predator populations will never be extinct. The equilibrium having zero predator population and non-zero prey population will always exist, and it will be globally stable if the maximum prey population inside the refuge is greater than . Otherwise it will be unstable. The co-existing equilibrium will be asymptotically stable if 1 ̅̅̅ is bigger than half of the carrying capacity, and otherwise will be unstable.
To bring the population to the desired level and to protect it from extinction, we use z-control, where the population reaches the desired level in a short time. We performed a simulation where the desired state was limit cycles instead of an equilibrium point. Numerically it is shown that as the intensity of fear increases, the population of the prey in refuge increases, while the population of the predator decreases i.e. fear is not in the favor of predator populations. To make our study more ecologically realistic, we took different forms of refuge size ( ) i.e. linear, exponential, and logistic instead of constant. We observed that when refuge size is linear or exponential, the prey out of the refuge and in the refuge tend to attend their stable equilibrium and predators go to extinction. If we consider refuge size as a logistic function of the prey out of refuge, then after a little oscillation all three populations co-exist. The adaptive control inputs for asymptotic stability are obtained as non-linear feedback. We examined the stability of the system with and without control and noted that the system with control approaches stability faster than the system without control.

Conflict of interest
The authors declare no conflict of interest.