STABILITY OF EIGENVALUES FOR PREDATOR- PREY RELATIONSHIPS

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Research Article STABILITY OF EIGENVALUES FOR PREDATOR- PREY RELATIONSHIPS Tailor Ravi M., 2 Bhathawala P.H. Address for Correspondence Assistant professor of Laxmi Institute of Technology, Sarigam, Valsad 2 Principal of S.

1 Research Article STABILITY OF EIGENVALUES FOR PREDATOR- PREY RELATIONSHIPS Tailor Ravi M., 2 Bhathawala P.H. Address for Correspondence Assistant professor of Laxmi Institute of Technology, Sarigam, Valsad 2 Principal of S.S. Agrawal Institute of Management and Technology, Gandevi road, Navsari. E Mail ravi_just35@yahoo.co.in ABSTRACT In this paper, predator prey model for scientific problem has been formulated. Shark predator-prey model for the differential equations is defined. In order to check the system s stability, eigenvalues are required for that a set of equilibriums points are discussed. KEYWORDS: Lotka-Volterra Model, Equilibrium, Jacobian matrix, Stability, Eigen values. INTRODUCTION For the purpose of describing the interaction between predators and prey, we decided to imagine the relationship between Sharks and. In this model, F will always represent the population of the Fish and S will represent the population of the Sharks. The population growth of the Fish is dependent on Sharks and of natural causes. The population growth of the Sharks is dependent on the number of Sharks that die from natural causes. Formulation of the Scientific Problem: There are many instances in nature where one species of animal feeds on another species of animal, which in turn feeds on other things. The first species is called the predator and the second is called the prey. Theoretically, the predator can destroy all the prey so that the latter become extinct. However, if this happens the predator will also become extinct since, as we assume, it depends on the prey for its existence. What actually happens in nature is that a cycle develops where at some the prey may be abundant and the predators few. Because of the abundance of prey, the predator population grows and reduces the population of prey. This results in a reduction of predators and consequent increase of prey and the cycle continues. An important problem of ecology, the science which studies the interrelationships of organisms and their environment, is to investigate the question of coexistence of the two species. To this end, it is natural to seek a mathematical formulation of this predator-prey problem and to use it to forecast the behaviour of populations of various species at different s. Risk and Food Availability: Sharks appear to be a major threat to. Availability of prey helps animals decide where to live. Predator-Prey Model: Fish & Sharks: We will create a mathematical model which describes the relationship between predator and prey in the ocean. Where the predators are and the prey are. In order for this model to work we must first make a few assumptions. Assumptions: Fish only die by being eaten by, and of natural causes. Shark only die from natural causes. The interaction between shark and can be described by a function. Differential Equations and how it relates to Predator-Prey: One of the most interesting applications of systems of differential equations is the predator-prey problem. In this project we will consider an environment containing two related populations-a prey population, such as, and a predator population, such as. Clearly, it is reasonable to expect that the two populations react in such a way as to influence each other s size. The differential equations are very much helpful in many areas of science. But most of interesting real life problems involves more than one unknown function. Therefore, the use of system of differential equations is very useful. Without loss of generality, we will IJAET/Vol.II/ Issue II/April-June, 2/4-49

2 concentrate on systems of two differential equations. The Lotka Volterra Model: The Model The growth rate of the population is influenced, according to the first differential equation, by three different terms. It is positively influenced by the current population size, as shown by the term, where a is a constant, nonnegative real number and is the birth rate of the. It is negatively influenced by the natural death rate of the, as shown by the term, where b is a constant, nonnegative real number and is the natural death rate of the. It is also negatively influenced by the death rate of the due to consumption by as shown by the term, where System is a constant non-negative real number and is the death rate of the due to consumption by. The Model Initial Conditions F : The population of the at t. S : The population of the at t. : The initial size of the population. : The initial size of the shark population. a : Reproduction rate of. b : Death rate of. C : Proportional to the number of that a shark can eat. D : Amount of energy that a supplies to the consuming shark. K : Death rate of., the growth rate of the Shark population, is influenced, according to the second differential equation, by two different terms. It is negatively influenced by the current shark population size as shown by the term, where is a constant nonnegative real number and is the shark population. It is positively influenced by the shark interactions as shown by the term, where d is a constant nonnegative real number, is the shark population and is the population. Equilibrium Points: Once the initial equations are understood, the next step is to find the equilibrium points. These equilibrium points represent points on the graph of the function which are significant. These are shown by the following computations. Let and To compute the equilibrium points we solve Solution: When and When Solution:. Now we find all the combinations: One of our equilibrium points is. or or For When, then Thus, one of our equilibrium points is. For and ; Solution: Thus, one of our equilibrium points is Our equilibrium points are Now, to study the stability of the equilibrium points we first need to find the Jacobian matrix which is:,. IJAET/Vol.II/ Issue II/April-June, 2/4-49

3 To study the stability of :, Solution: Semi-stable since one eigenvalue is negative and one is positive. To study the stability of :, Solution:. Stable if. Semi-stable if. To study the stability of : J = Solution: If we simplify a little more, we get: Stable since both of the real parts are negative. The imaginary numbers tells us that it will be periodic.. Different Case Study For Checking The Variation In Eigen Value Case :, INPUT function ydot = subprog(t,y); yd()=6*y()-2*y()^2-4*y()*y(2); yd(2)=-3*y(2)+5*y()*y(2); ydot = [yd() yd(2)]'; subprog3 x= y=.5 [t y]=ode45(@subprog,[ 5],[x y]); plot(y) legend('',' ') xlabel('') ylabel(',') IJAET/Vol.II/ Issue II/April-June, 2/4-49

4 OUTPUT ,s harks , , IJAET/Vol.II/ Issue II/April-June, 2/4-49

5 In this case, of course is the critical point for the population in a shark-free world, and for the population living with. This case holds true regardless of the initial conditions, as long as, and. Case 2:, INPUT function ydot = subprog(t, y); yd()=2*y()-6*y()^2-4*y()*y(2); yd(2)=-3*y(2)+5*y()*y(2); ydot=[yd() yd(2)]'; subprog3 x= y=.5 plot(y) legend('',' ') xlabel('') ylabel(',').2 ;.8 fis h,s hark s OUTPUT fis h,s hark s IJAET/Vol.II/ Issue II/April-June, 2/4-49

6 .6.5.4, In this case, when is less than conclusion, which is supported by simple algebra. When, then for the shark equation, the critical point becomes a negative number! results in a negative number. Therefore in this case, the shark population will die out REGARDLESS of the initial conditions! So the solution would converge to the shark-free stable point. Case 3: All constants are equal, INPUT function ydot=subprog(t,y); yd()=*y()-*y()^2-*y()*y(2); yd(2)=-*y(2)+*y()*y(2); ydot=[yd() yd(2)]'; subprog3 x= y=.5 [t y]=ode45(@subprog,[ 5],[x y]); plot(y) legend('',' ') xlabel('') ylabel(',') OUTPUT.9, IJAET/Vol.II/ Issue II/April-June, 2/4-49

7 .5, , In this case, what if all of the constants were the same? A simple glance at the equations tells us that this would be similar to the second case: We get, yet the critical point would be which is on the y-axis (S) and identical to the stable point for the population in a shark-free world. So here the die out again. (But it s hard to feel sorry for!) Case4:( ),. INPUT function ydot=subprog(t,y); yd()=2*y()-*y()^2-*y()*y(2); yd(2)=-*y(2)+*y()*y(2); ydot=[yd() yd(2)]'; subprog3 x=2 y=3 [t y]=ode45(@subprog,[ 5],[x y]); plot(y) legend('',' ') xlabel('') ylabel(',') IJAET/Vol.II/ Issue II/April-June, 2/4-49

8 OUTPUT , , in this case, we make which turns the model into the simplest form of the predator/prey model. The new equations look like this: and So the critical point becomes and we get an ellipse around the critical point, the shape and size depending on the constants and initial conditions. So both the and shark populations wax and wane in a cyclic pattern with the lagging behind the. Case 5: INPUT function ydot=subprog(t,y); yd()=2*y()-*y()^2-*y()*y(2); yd(2)=-*y(2)+*y()*y(2); ydot=[yd() yd(2)]'; subprog3 x= IJAET/Vol.II/ Issue II/April-June, 2/4-49

9 y=.5 [t 5],[x y]); plot(y) legend('',' ') xlabel('') label(',') , OUTPUT T , Now for the last case, we pose the question: What happens when, i.e.? Answer: This is pretty much similar to the first case, since, with a simpler spiral as the result. The only significant impact is the location of the critical point. There are other cases that we have yet to explore here, such as a=, c=, k =,, or a combination of those, but those would render the model meaningless, as λ they would cancel the relationship between the and the or eliminate the s growth rate or the shark s death rate. IJAET/Vol.II/ Issue II/April-June, 2/4-49

10 CONCLUSION This Lotka-Volterra Predator-Prey Model is a rudimentary model of the complex ecology of this world. It assumes just one prey for the predator, and vice versa. It also assumes no outside influences like disease, changing conditions, pollution, and so on. However, the model can be expanded to include other variables, and we have Lotka-Volterra Competition Model, which models two competing species and the resources that they need to survive. We can polish the equations by adding more variables and get a better picture of the ecology. But with more variables, the model becomes more complex and would require more brains or computer resources. This model is an excellent tool to teach the principles involved in ecology, and to show some rather counter-initiative results. It also shows a special relationship between biology and mathematics. Now, what does this has to do with orbital mechanics? Simple: this model is similar to the models of orbits with those spirals, contours and curves. We can apply this model with constants representing gravitational pulls and speeds of bodies. Nomenclature: F : The population of the at t. S : The population of the at t. a : Reproduction rate of. b : Death rate of. C : Proportional to the number of that a shark can eat. D : Amount of energy that a supplies to the consuming shark. K : Death rate of. Superscript: : Initial condition REFERENCES. Boyce, William. Elementary Differential Equations. New York: John Wiley & Sons. 2. Cullen, Michael & Zill, Dennis. Differential Equations with Boundary-Value Problems. 3. Zill, Dennis. A First Course in Differential Equations: The Classic Fifth Edition. 4. Numerical Linear Algebra And Applications by Biswa Nath Datta IJAET/Vol.II/ Issue II/April-June, 2/4-49

A NUMERICAL STUDY ON PREDATOR PREY MODEL

A NUMERICAL STUDY ON PREDATOR PREY MODEL International Conference Mathematical and Computational Biology 2011 International Journal of Modern Physics: Conference Series Vol. 9 (2012) 347 353 World Scientific Publishing Company DOI: 10.1142/S2010194512005417