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الباحثة رحاب نوري On the dynamics of some Prey-predator Models with Holling type-IV functional response

Abstract This thesis deals with the dynamical behavior of prey-predator models with nonmonotonic functional response. Two types of prey-predator models are proposed and studied. The first proposed models consisting of Holling type-IV prey-predator model involving intra-specific competition

However, the second proposed models consisting of Holling type-IV prey-predator model involving intra-specific competition in prey population with stage structure in predator. Both the models are represented mathematically by nonlinear differential equations. The existence, uniqueness and boundedness of the solution of these two models are investigated. The local and global stability conditions of all possible equilibrium points are established.

The occurrence of local bifurcation (such as saddle-node, transcritical and pitchfork) and Hopf bifurcation near each of the equilibrium points are discussed. Finally, numerical simulation is used to study the global dynamics of both the models.

Chapter one Preliminary

Mathematical model; is mathematical construct, often an equation or a relation ship, designed to study the behavior of a real phenomena. Definition: Any group of individuals, usually of a single species, occupying a given area at the same time is known as population. The application of mathematical method to problem in ecology has resulted in a branch of ecology know as mathematical ecology,. There are three different kinds of interaction between any given pair of species, these are described as bellow.

mutualisms or symbiosis (+,+), in which the existence of each species effects positively on the growth of the other species (i.e. increases in the first species causes increases in the growth of the second species ) Competition (-,-), in which the existence of each species effect negatively on the growth of the other species. prey-predator (-,+) in which the existence of the first species (prey) effects positively (increases) on the growth of the second species (predator), while the existence of the second species

(predator) effects negatively (decreases) on the growth of the first species (prey). Functional response; is that function, which refers to the change in the density of prey attacked per unit at time per predator as the prey density changes .the functional response can be classified into three classes with respect to its dependence on the prey or predator ,those are : prey-dependent functions : the term prey-dependent means that the consumption rate by each predator is a function of prey density only, that is . In the following, the most commonly used prey-dependant types functional reasons are presented

(i) lotka-volterra type: in which the consumption rat of individual predator increases linearly with prey. Hence the lotka - volterra type of functional response can be written as follows: Where is the consumption rate of prey by p rate of prey by predator. (1) (1)

(ii) Holling type-I In which the consumption rate of individual predator increases lineally with prey density and when the predator satiate then it becomes constant. Hence the Holling type-I functional response can be written as follows: Where is the value of prey density at which the predator satiate at constant

(iii) Holling type-II: in which the consumption rate of individual predator increases at a decreasing rate with prey density until it becomes constant at satiation level. It is a hyperbola the maximal value asymptotically, and defined by Here is a search rate, h is the time spent on the handling of one prey, is the maximum attack rate and b is the half saturation level (3)

(iv) Holling type-III: In this type of functional response the consumption rate of individual predator accelerates at first and then decelerates towards satiation level. It is defined as: (4)

(V) Holling type-IV: The type of functional response suggested by Andrews. The Holling type IV functional response is of the form. (5)

2. Ratio-dependent function : The prey-dependent functional responses do not incorporate the predator abundance in it forms. Therefore, Aridity and Ginsberg suggested using new type of functional response called ratio-dependent, in which the rate of prey consumption per predator depends not only on prey density but instead depends on the ratio between the prey and predator density. The Aridity -Ginsberg ratio dependent response can be written as: (6)

3.predator-dependent functions; In this type, the functional responses are depending on both prey and predator densities. It is observed that high predator density leads to more frequent encounters between predators. The most commonly used predator dependent functional response is that proposed by DeAngalies and independently by Beddingnon, which is now know as DeAnglis-Beddington functional response: (7)

Chapter two The dynamics of Holling type IV prey predator model with intra- specific competition

Mathematical models Let x be the density of prey species at time ,y be the density of predator species at time that consumes the prey species according to Holling type IV functional response then the dynamics of a prey–predator model can be represented by the following system of ordinary differential equation. (8)

with and Note that all the parameters of system (8) are assumed to be positive constants and can be described as follow: is the intrinsic growth rate of the prey population; is the intrinsic death rate of the predator population; the parameter is the strength of intra-specific competition among the prey species; the parameter can be interpreted as the half-saturation constant in the absence of any inhibitory effect; the parameter is a direct measure of the predator immunity from the prey; is the maximum attack rate of the prey by a predator; represents the conversion rate; and finally is the strength of intra-specific competition among the predator species .

Now I worked on my research and I finished to finding and proofing the following 1. Existence and Local Stability analysis of system (8) with persistence (1) The trivial equilibrium point always exists. (2) The equilibrium point always exists, as the prey population grows to the carrying capacity in the absence of predation.

(3) There is no equilibrium point on as the predator population dies in the absence of its prey. (4) The positive equilibrium point exists in the interior of the first quadrant if and only if there is a positive solution to the following algebraic nonlinear equations: We have the polynomial from five degrees

And its exists under the condition, and when I investigated varitional matrix for my system and substitute we get the eigenvalues are and Therefore, is locally asymptotically stable if and only if

While is saddle point provided that And when I investigated varitional matrix for my system and substitute we get is locally asymptotically stable if and only if and

3.The System(8) has a Hopf –bifurcation. And the persistence is proof by using the Gard and Hallam method. 2.Global dynamical behavior of the system (8) I proof the global dynamical behavior of system (8) and investigated by using the Lyapunov method 3.The System(8) has a Hopf –bifurcation. Near the positive equilibrium point at the parameter value 4.The system (8) is uniformly bounded

5. Numerical analysis (8a) In this section the global dynamics of system (8) is studied numerically. The system (8) is solved numerically for different sets of parameters and for different sets of initial condition, by using six order Runge-Kutta method (8a)

Fig1: (a) Globally asymptotically stable positive point of system (8) for the data starting from (8a) with different initial values. (b) Time series of the attractor given by Fig 1(a)

Fig. 2: (a) The system (8) approaches asymptotically to stable point with the data given in (8a) with starting from (3, 3), (b) Time series of the attractor given by Fig1 (a).

Fig2: Globally asymptotically stable limit cycle of system (8) for the data given by Eq. (8a) with (a) The solution approaches to limit cycle from inside (b) The solution approaches to limit cycle from outside (c) Time series for Fig (1a) (d) Time series for Fig (1b).

Chapter three Stability Analysis of a stage structure prey-predator model with Holling type IV functional response

1.The mathematical models

(2a) The trivial equilibrium point always exists. 1.All the solutions of system (9) which initiate in are uniformly bounded 2. Existence and Local Stability analysis of system (9) are investigate (2a) The trivial equilibrium point always exists. (2b) It is well known that, the prey population grows to the carrying capacity in the absence of predator, while the predator population dies in the absence of the prey, then the axial equilibrium point always exists.

(4) The positive equilibrium point (3) There is no equilibrium point in the as the predator population dies in the absence of its prey. (4) The positive equilibrium point exists in the Now we obtain the following third order polynomial equation.

5.Numerical analysis 3. Global dynamical behavior of system (9) are investigate with the help of Lyapunov function 4.The Bifurcation of system (9) the occurrence of a simple Hopf bifurcation and local bifurcation (such as saddle node,pitck fork and a transcritical bifurcation) near the equilibrium points of system (9) are investigated. 5.Numerical analysis In this section the global dynamics of system (9) is studied numerically. The system (9) is solved numerically for different sets of parameters and for different sets of initial condition, by using six order Runge-Kutta method

(9a) Fig. (1): (a) The solution of system (9) approaches asymptotically to the positive equilibrium point starting from different initial values for the data given by Eq. (9a). (b) Time series of the attractor in (a) starting at (0.85, 0.75, 0.65).

(c) Time series of the attractor in (a) Fig. (2): (a) Globally asymptotically stable limit cycle of system (9) starting from different initial values for the data given by Eq. (9a) with (b) Time series of the attractor in (a) starting at (0.85, 0.75, 0.65). (c) Time series of the attractor in (a) (1.25, 0.25, 0.75). starting at Fig. (3): The trajectory of system (9) approaches asymptotically to stable point for with the rest of parameter as in Eq. (9a).

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