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CR MATH 55 - Predator-Prey Modelling

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IntroductionExamining The DataBuilding A ModelAnalyzing Model 1A Second ModelA Further RefinementLinear PredationA Better Predation ModelA New Predator ModelAnalyzing Model 3Final ThoughtsIntroductionExamining The DataBuilding A ModelAnalyzing Model 1A Second ModelA Further RefinementA New Predator ModelAnalyzing Model 3Final ThoughtsHome PageTitle PageJJ IIJ IPage 1 of 27Go BackFull ScreenCloseQuitPredator-Prey ModellingEric P. AnopolskyMay 21, 2004AbstractNonlinear analysis is applied to three different predator-prey models to determine the behaviorof solutions. For each system of differential equations, equilibrium points are found and examined.Linearizations are constructed for interesting equilibrium points and classified based on the param-eters of the system. Changes to some parameters in some systems do not change the fundamentalbehavior of solution curves in the phase plane, while others can change stable equilibrium pointsinto unstable points and vice versa.1. IntroductionAs long as there is interest in studying living things, there will be a need to model populations math-ematically, specifically sets of populations with predator-prey interactions. They can be used to maketestable predictions about the behavior of a number of populations, and provide clues as to why theybehave the way they do. Sometimes it is necessary to go a step further and manipulate populations, butdoing so requires knowing what can be manipulated, how, when, and what the results will be. Whichset of mathematical tools are the right ones for the application? The answer comes from the data.2. Examining The DataSupp ose the following tables contain the population data for two populations in isolation.IntroductionExamining The DataBuilding A ModelAnalyzing Model 1A Second ModelA Further RefinementA New Predator ModelAnalyzing Model 3Final ThoughtsHome PageTitle PageJJ IIJ IPage 2 of 27Go BackFull ScreenCloseQuitYear Population 1 Population 22000 10000 50002001 20000 100002002 40000 200002003 80000 400002004 160000 80000A number of observations could be made about this data. One is that both populations are clearlyexponential. Population 1 seems to obey the equation P = 10000 · 2t−2000where P is the populationand t is the year. Population 2, similarly, seems to fit the equation P = 5000 · 2t−2000.Another more interesting observation is that both population growth rates seem to be proportionalto the size of the population. Mathematically, P0= rP , where the P0is dP/dt (the growth rate of thepopulation) and r is some constant. This is known as the Malthusian model, and it is a differentialequation. A different way of expressing this relationship is P0/P = r. In other words, the growth rateof the population per individual is constant.The second representation of the model makes it clear that the equation is separable. Solving itproceeds as follows.P0P= rZ1PdP =Zrdtln |P | = rt + CNegative population values do not make sense, soln P = rt + CP = ert+CP = eCertP = Aert.To find a particular equation representing either Population 1 or Population 2 from above, it would benecessary to find particular values for the constants A and r, which is relatively easy.This example provides two important pieces of information. The first is that populations can bemodelled quite well by differential equations. More than one population, then, would require more thanIntroductionExamining The DataBuilding A ModelAnalyzing Model 1A Second ModelA Further RefinementA New Predator ModelAnalyzing Model 3Final ThoughtsHome PageTitle PageJJ IIJ IPage 3 of 27Go BackFull ScreenCloseQuitone differential equation — a system of differential equations. Two noninteracting species might bemodelled by a system likeA0= f(A)B0= g(B)(1)The second important piece of information is that finding an equation describing a particular popu-lation means finding a solution to the differential equation (or system of differential equations). As theDEs used to model populations become more and more complicated, this will not always be possible.Also, it will be impossible to consider each equation independently as systems will generally be of theform:A0= f(A, B)B0= g(A, B)(2)instead of the form of system (1).3. Building A ModelOne simple system that is both nonlinear and models interaction (like system (2)) isH0= rH − dHPP0= −sP + fHP.(3)This system models two populations: H (herbivores, the prey species) and P (predators). The pa-rameters r, d, s, and f are positive. When there are no predators, the herbivore equation simplifiesto H0= rH, the malthusian model for a single population from earlier. The natural growth rate ris greater than 0 so the population grows when it is the only population inhabiting an area. In theabsence of prey, the predator equation simplifies to P0= −sP . The natural growth rate −s is lessthan 0 so the predator population shrinks in absence of food. HP is the number of herbivore-predatorinteractions (more of each population means more interactions population). Some positive portion ofthose interactions, dHP decreases the growth rate of the prey s pecies, and some other positive portionof those interactions, f HP increases the predator species growth rate.Another way of looking at these systems is to divide the prey e quation through by the number of preyand the predator equation by the number of predators. This requires, of course, that neither populationIntroductionExamining The DataBuilding A ModelAnalyzing Model 1A Second ModelA Further RefinementA New Predator ModelAnalyzing Model 3Final ThoughtsHome PageTitle PageJJ IIJ IPage 4 of 27Go BackFull ScreenCloseQuitbe 0.H0H= r − dPP0P= −s + fH(4)The reproduction rate of the average prey is some constant r, but as more predators are added to theenvironment the reproduction rate decreases until it is 0 and then becomes more and more negativewith the continued increase in predator density. Similarly, when there are no prey, the “reproduction”rate of the average predator is negative – it will starve. When there are sufficient prey to sustain thatpredator but no additional offspring, it will not produce any. When there are more than enough prey,the predators will produce increasing offspring with increasing food resources.4. Analyzing Model 1One tool from nonlinear analysis that can be applied to system (3) is examining the linearization aroundan e quilibrium point using the Jacobian. In order to do this, we need to find the Jacobian for the systemand plug in


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