BIENVENUE A LA SOUTENANCE DE MASTER DE: Soh guy blondeau VEUILLEZ GARDER LE CALME ET ETEINDRE VOS TELEPHONES PORTABLES S.V.P. 1

REPUBLIQUE DU CAMEROUN ************ Paix-Travail-Patrie ************ UNIVERSITE DE DSCHANG ************ ECOLE DOCTORALE ************ UFD SCIENCES FONDAMENTALES ET TECHNOLOGIE REPUBLIC OF CAMEROON ************ Peace-Work-Fatherland ************ UNIVERSITY OF DSCHANG ************ POST-GRADUATE COLLEGE ************ UFD FUNDAMENTAL SCIENCE AND TECHNOLOGY UNDER THE DIRECTION OF Dr. TCHITNGA ROBERT Lecturer at the University of DSCHANG March 2016 Presented by : SOH GUY BLONDEAU Bachelor degree in physics Registration Number: CM04-10SCI0909 APPLICATION OF RELAY TECHNICS TO PREY-PREDATOR DYNAMICS 2

IV CONTROLLED AREA WITH REFUGE: REFORMATION OF PREY AND PREDATORS III CONTROLLED AREA WITH REFUGE: REFORMATION OF PREY II CONTROLLED AREA WITH REFUGE PRACTICAL STABILTY I CONTROLLED AREA WITHOUT REFUGE CONCLUSION AND PERSPECTIVES INTRODUCTION 3

4 over predation, restriction on harvesting, bush fire environmental pollution, etc. creating reserve zones/refuges

Type I Type II Type III FIG. 1.1: Graphical representation of all three Holling predator functional response. MATHEMATICAL MODELS Holling type I, II, and III functional responses. 5

Relay unit Ki Kj Holling type II 6 Refuge X 1 (t ) X 2 (t ) z 1 (t ) z 2 (t ) y 1 (t ) y 2 (t )

CONTROLLED AREA WITHOUT REFUGE I 7  Two uncontrolled forest x(t) and z(t)  The controlled area y(t) FIG. 1.2: Schematic representation of the connected regions in case without refuge Refuge

8 a 1 is the intrinsic growth rate of the prey (a1>0) b1 is the death rate of the predator (b1<0) C1 and C2 are the control parameters K1, K2 and K3 are the coupling parameters a 2 et b 2 are the intra-species competition rate a 2, b 2 >0

The research center is built such that the particular new system obtained through the following expression The controlled area model be written as follows: 9

Stability of the system without the refuge According to LYAPONOV theorem ζ‘s system is asymptotically stable and indirectly the entire network. Let us consider the following Lyapunov function: The derivative is given by

FIG. 1.4: Synchronization between preys and predator densities with C 1 = C 2 =0.1 11

Numerical results FIG. 1.3: Time behaviors of prey and predator densities with C1= C2=0.1 12

The predator’s growth can disturbed the stability of our system by multiplied attack. To protect the prey and preserved the stability we need to incorporate a refuge into the controlled area

CONTROLLED AREA WITH REFUGE: PRACTICAL STABILTY II 14 FIG. 2.1: Schematic representation of the connected regions in the presence of refuge Taking into consideration that the refuge the refuge protects m x of the prey from predator, the association refuge and Controlled Area is modeled as follow: Refuge

2.1 15

FIG. 2.2: Behaviors of preys densities inside the refuge with K4=0.16 and K5= FIG. 2.3: Behaviors of preys densities inside the refuge with K4= and K5= However, increasing the amount of refuge can increase prey densities and lead to population outbreaks. To avoid it, we must reduce a quantity of prey in the refuge to give it in the starting system: it is the reformation.

17 CONTROLLED AREA WITH REFUGE: REFORMATION OF PREY III FIG. 3.1: Representation of the reformation between the connected regions (Black arrow) Refuge

Where g(t) is the sitting apart rate function from the refuge

19 the association refuge and Controlled Area is modeled as follow: Then the rest of the previous deduction which is send to elsewhere is given by the relation:

20 Numerical results FIG. 3.3: Behaviors of preys densities inside the refuge with

21 FIG. 3.4: Time behaviors of predator densities withFIG. 3.5: Time behaviors of predator densities with

22 CONTROLLED AREA WITH REFUGE: REFORMATION OF PREY AND PREDATORS IV FIG. 4.1: Representation of the reformation between the connected regions (Black arrow) The distribution of the preys taken into the refuge is represented with black arrows while the repartition of the predators is represented in red arrows Refuge

The predator differential equation of the controlled forest is modeled as follow: Then the rest of the previous deduction which is sent elsewhere is given by the relation:

25 FIG. 4.3: Behaviors of preys densities inside the refuge with FIG. 4.2: Behaviors of preys densities inside the refuge with

26 CONCLUSION IV Refuge plays a very important role to ensure either existence of prey or their extinction Reformation can help to fight against extinction of species taking into account the necessity of production of meat for human populations and the exportations of species

27 Electronic device Delay Climatic changing Holling type III

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