Diferencia entre revisiones de «Predator-Prey Model (grupo 3)»

Revisión del 20:37 4 mar 2013

1 INTRODUCTION

The Lotka – Volterra equations, also known by the name of predator-prey equations, are a pair of first order and non linear differential equations. They are commonly used to describe the model in which two species (predator and prey) interact one with the other, their interactions and competitions. dR/dt= aR– bRF dF/dt= −cF+ dRF

2 INTERPRETATION

First of all,we are going to describe the physical meanings term by term: - R is the number of prey, in this case, Rabbits. - F is the number of predators, for this example, theFoxes. - dR/dt is the growth of the population of Rabbits. - dR/dt is the growth of the population of Foxes. - a, b, c and d are parameters describing the interaction of the species. Now we aregoing to analyze how they are related with themselves. - First we assume that thepreys have an unlimited food supply. The increase of rabbits, in absence offoxes (F=0) is proportional to the initial number of rabbits. The growth of therabbits is affected by the interaction between the rabbits and foxes (FR). The“b” parameter indicates this process. So we can conclude that the change inrabbitpopulation is given by its growth minus the number of eaten by the foxes. dR/dt= aR– bRF - We have to assume that thefood supply for the foxes depends entirely on the rabbit population. The“-c”parameter tell us about the decreasing rhythm of the foxes, affected by theirown death, emigration, etc. In absence of prey (R=0), it leads to anexponential decay.

dF/dt= −cF+ dRF

3 NUMERICAL RESOLUTION

We have to apply iterative methods to solve the equations, because weare in the case that the equations are non linear. NOTE: at allthe exercises we use the scale 1:5000.

3.1 EULER METHOD

For an initial population of R=3000 and F=1000, using an interval of time(0,100) and using a=0.4, b=0.37, c=0.3, d=0.05. The OCTAVE code for 5000intervals is:

The graphic solutions are:

We have taken 5000iterations trying to obtain the realest solution (this is because, according tothe Euler Method the higher the number of N is, the shorter will be thedistance h and closer to the real solution it will be. The system solution is cyclical. We can affirm that theecosystem is not stable because, as we can see clearly in the graphics, as weadvance in time, there are more differences between foxes and rabbits(sometimes even the population of foxes is bigger than the rabbit’s), so thesystem is less stable.

Now we are going to suppose three cases with three different values of rabbit’spopulation.

As we can see,in all the graphics the number of rabbits is growing up among time. Logically,the less rabbits there are, more foxes will die at the beginning, and this willcause that the number of rabbits will reach more maximums. The number offoxes will reach the lowest minimums when the initial population of rabbits isthe lowest, and as a consequence of this, when the number of rabbits reach itsmaximum, the population of foxes will grow up again to reach its maximum. Talking aboutthe proximity to the limit of foxes before extinction (50), in the last case italmost reach this point, getting very close and getting into a dangerous pointto these animal.

3.2 RUNGE-KUTTA METHOD