Predator-Prey Model with Ratio-dependent Food

University of Minnesota Duluth Department of Mathematics and Statistics Predator-Prey Model with Ratio-dependent Food Processing Response Advisor: Harlan Stech Jana Hurkova June 2013

Table of Contents 1. Introduction... 1 1.1. Known models... 1 1.2. Functional response... 1 1.3. Rosenzweig-MacArthur model... 4 2. Model Development... 7 3. Rescaling... 10 4. Nullclines... 11 4.1. Prey nullcline... 12 4.2. Predator nullcline... 16 4.3. Saddle node bifurcation... 18 4.4. Transcritical bifurcation... 20 5. Equilibria... 25 5.1. Extinction... 25 5.2. Monoculture equilibrium...26 5.3. Coexistence equilibrium... 26 5.4. Hopf bifurcation... 38 5.5. Solution... 43 5.6. Homoclinic bifurcation... 45 6. Comparison to Rosenzweig-MacArthur Model... 45 7. Suggestions for Future Research... 46 Table of figures... 47 References... 48 Appendix A - Saddle node bifurcation Appendix B - Phase plane plots Appendix C - Hopf bifurcation

Acknowledgments I would like to express the deepest appreciation to my committee chair, Professor Harlan Stech, who has been the best advisor and professor I have ever had. Thank you for your guidance, patience, valuable advices and grammar checks. You introduced me the interesting side of mathematical modeling. I would like to also thank my committee members, Professor Bruce Peckham and Professor John Pastor for serving on my committee. In Addition, a thank you to Professor Dalibor Froncek, whose friendly approach makes all students feel home. You are a big support and the way you care about all students is amazing.

Abstract Predator-Prey models have been studied for a long time. The most famous ones are the Lotka- Volterra model, the Lotka-Volterra model with logistic function, and the Rosenzweig-MacArthur model. The properties of these models are well known. The purpose of this paper is to introduce a new model based on the Rosenzweig-MacArthur model using a new functional response that depends on the ratio of the predator and the prey populations. The model is analyzed and simulated, and its properties compared to the Rosenzweig-MacArthur model. The new model shows a surprising variety of behaviors, including multiple coexistence steady states, simultaneously attracting periodic and steady states, and homoclinic loss of periodic solution.

1. Introduction 1.1. Known models The predator-prey models, also known as producer-consumer models, describe the dynamics of biological systems in which two species interact, one a predator and one its prey. The simplest predator-prey model is the Lotka-Volterra model. This model was proposed in 1925 by the American biophysicist Alfred Lotka and the Italian mathematician Vito Volterra. The assumptions of this model are: density-independent exponential growth of the prey population in the absence of predators, a constant predation rate, a constant conversion rate and a constant per capita mortality rate of predators. Unfortunately, this model does not describe actual behavior observed in nature. One of the biggest problems is that the prey population is not self-limiting and, therefore, it can be arbitrary large. In order to fix this problem, a new version of the Lotka-Volterra model was introduced in 1930. This model uses a logistic growth instead of exponential growth for the prey. The logistic growth guarantees that the prey population is self-limiting. The prey can grow only to a certain saturation level. See for example (Pastor 2008) for more details. Our paper introduces a model related to the Rosenzweig- MacArthur model, which was proposed in 1963 in American Naturalist. See (Rosenzweig & MacArthur, 1963). The difference between the Lotka-Volterra model with logistic growth and the Rosenzweig-MacArthur model is that the predation rate is no longer assumed to be proportional to prey density. 1.2. Functional response There are many examples in nature which demonstrate that predators control the numbers of their prey. When predators are faced with increasing local density of their prey, they often respond by changing their consumption rate. This relationship of an individual predator's rate of food consumption to prey density was termed the "functional response" by C. S. Holling in 1959. He analyzed the act of predation and broke it onto behavioral units termed the components of 1

predation. There are many factors that might influence predation but we will consider only four major components: search, capture, handling and digestion. Theory and laboratory testing suggest that the functional response of an animal may take one of three forms called the Type I, Type II and Type III functional responses. See (Real, 1977), (Kuang, 2007), (Liu & Lou, 2010) or (Cosner, DeAngelis, Ault, & Olson, 1999). Figure 1: Three types of functional response Type I is the simplest functional response in which prey consumption rises linearly with prey density to a threshold level. This is called a filter feeder's functional response. However, increasing prey density beyond a threshold does not result in an increased per capita predation rate. This is due to digestion and handling components of predation. Figure 2: Type I functional response 2

The Type II functional response is a cyrtoid functional response in which the attack rate increases at a decreasing rate with prey density until it becomes constant at satiation. Cyrtoid behavioral responses are typical of predators that specialize on one or a few prey. Figure 3: Type II functional response In the Type III response, prey consumption remains low until a threshold density is reached. The predation rate then increases exponentially until levels out. The shape of this curve is described as sigmoidal. The predator ignores the prey when densities are very low in order to make hunting energetically profitable. Figure 4: Type III functional response 3

1.3. Rosenzweig-MacArthur model The Rosenzweig-MacArthur model uses the Holling II type functional response. See (Real, 1977). It belongs to the category of models with Predator Food Handling Times. After killing a prey, predator eats its captured food. This process consists of eating and digesting the meal portion. This fact was not considered in previous models (Lotka-Volterra) in which the predation rate was linearly proportional to prey density. The Rosenzweig-MacArthur model is inspired by behavior that can be found in nature. We cannot assume that the predator eats its prey at a constant rate. Some complications might occur, for example not having enough prey to hunt, delay in digesting, etc. The functional response in general is where interval, and is time interval necessary for a predator to catch its prey, also called the encounter is a time interval for chewing and digesting the captured prey. The Holling II type functional response (predation rate) is After using a substitution and, the functional response has the form 4

where is called the predation half-saturation constant and it is the density of prey at which the predator's killing rate reaches half its maximum value,. The aggregated model is where is a birth/death rate, is the carrying capacity, is the maximum per capita killing rate, is the value of that gives half the per capita killing rate (half-saturation constant), is the biomass conversion efficiency term that specify how much of the biomass of a killed individual is transformed into predator biomass. We assume that predators die with random death rate. Nullclines Nullclines are curves where the rate of change of the equations are 0, that is where or. The equilibria lie on intersection of these curves. There are two sets of nullclines. Predator nullclines: Prey nullclines: 5

The x-axis and y-axis are trivial nullclines. Non-trivial nullclines are a parabola and a vertical line. Figure 5: Rosenzweig-MacArthur model - nullclines Equilibria Three equilibria points can be found. Case I : This equilibrium describes the extinction of both species. Neither of the two species is alive. Case II : This equilibrium is called the monoculture equilibrium because only the prey population is alive, while predator population is extinct. This equilibrium is stable if, so in the case where predator nullcline is to the left of this equilibrium. It is unstable if, so when predator nullcline intersects the vertical (prey) nullcline. This behavior is called a transcritical bifurcation. Case III : Finally, this point is a coexistence equilibrium because both species are alive. The stability of this point also depends on values of. If the nullcline then lies to the right 6

of the maximum of the parabolic nullcline and the coexistence equilibrium is stable. Trajectories spiral into the coexistence equilibrium. If, the equilibrium is neutrally stable. If, the nullcline lies on the left of the maximum of the parabolic nullcline. The coexistence equilibrium is now unstable and trajectories spiral away from it. It can be shown by the Kolmogorv's theorem that the model contains either a stable equilibrium or a stable limit cycle, see (Pastor, 2008). It can be also shown by the Poincaré-Bendixson Theorem that if, the coexistence equilibrium is surrounded by a stable limit cycle. See (Hirsch, Smale, & Devaney, 2004), ( Kot, 2001) or (Rosen 1970 ). The change from stable to unstable coexistence equilibrium is called a Hopf bifurcation. 2. Model Development As mentioned before, the Rosenzweig-MacArthur model has a form This can be written more generally as where is predation term per capita (functional response). This term is determined by time intervals to catch and to process the prey 7

In the Rosenzweig-MacArthur model this term was The encounter rate is assumed to be proportional to prey density, therefore in this model This is the fact that we now focus on.. Also the handling time is assumed to be constant, so. Is it really always correct to assume that the time necessary to process food (chew, digest,...) is always constant in all situations that can occur in nature? Let us consider the situation where there is a pack of predators who hunts in a group. This behavior is very typical for many species, for example wolves, lions, hyenas, dogs, etc.. If there are more predators sharing a captured prey, the time to process will depend on the amount of food that each predator will get. Picture a scenario of wolves and a deer. Imagine a pack of 5 wolves hunting and eating a deer. After they finish eating their portions and resting they are ready to hunt again. Now imagine that there were 10 wolves instead. In that case each of them would get a smaller portion and it would be possible them to be still hungry. In that case they would be ready to hunt right away. Hunting in a group of individuals is a crucial factor that influence how soon predators are ready to hunt again. A slightly different situation with the same idea are rabbits (or cows) eating grass. The number of rabbits eating the grass in the same place influence the time they spend there and how long will it take them to move to a new area. Inspired by these natural situations, we introduce a new predation term where time interval to catch the prey stays the same, so we assume that the amount of time spend foraging does not depend on the size of the group, so 8

Therefore, the time to process food is not assumed constant, but it is assumed to be a saturating function of meal portion size. It can be noticed that when increases than the time to process decreases, and vice versa. With using this predation rate, the model we will study is where : is the density of prey(producer) population is the density of predator(consumer) population is the initial birth rate/death rate of prey is the carrying capacity for prey is the encounter rate is proportional to prey density is the maximum amount of food that each predator can eat is the half-saturation constant for the food processing time interval is the conversion efficiency constant is the death rate of predators. Predators are assumed to die for natural causes only. Note that if, then the model reduces to the Lotka-Volterra model. Note also that if then the model reduces to the Rosenzweig-MacArthur model. 9

3. Rescaling The new model includes 7 parameters. It is more efficient to decrease the number of parameters in order to analyze the model. The following rescaling was used. Let and. Then, and,. After substitution the system reads It is possible to simplify the model as follows After multiplying the 1st equation by and the 2nd equation by we obtain 10

We let, and, then the system becomes The number of parameters has been decreased to 5 but by change of time scale we can take and the number of parameters is reduced to 4. The parameter occurs only in new parameter. Therefore if we want to observe behavior of the system by increasing enrichment (increasing ), we need to vary only this one parameter. If we want to increase half-saturation to examine the effects of meal size, we will vary only the new parameter Similarly, varying predator death rate,, effects only the new parameter. 4. Nullcline analysis As usual, nullclines have been analyzed. Recall that nullclines are curves where a state variable derivative is zero. Hence, these are curves where or. Since is a common factor in the first equation of, it does not affect the nullclines for the system. In our case 11

There are 2 prey nullclines and 2 predator nullclines :. The x-axis and y-axis are zero nullclines. The non-zero nullclines are different than before. The predator nullcline is no longer a vertical straight line and the prey nullcline is no longer a parabola. The shapes of these curves are more complicated. 4.1. Predator nullcline ( ) Notice that the predator nullcline is dependent only on the two parameters and. By solving the equation for we obtain This curve is not a straight line anymore. It is not obvious what shapes can be expected. 12

Asymptotes: Since we assumed being positive, the curve has a vertical asymptote at. As the curve is asymptotic to a straight line with slope. x and y-intercepts: So the curve always passes through the origin However, if, so we are not interested in this branch. So, the predator nullcline intercepts the x-axis at the point. We are interested only in positive intercepts. Since we assumed all parameters are positive, the only way how this intercept can be positive is when, both conditions are satisfied when. We draw predator nullclines for all three cases; for and. 13

1) Figure 6: Predator nullcline, H=0.8, D=0.5 We can see that the curve crosses the x-axis at positive value of. Figure 7: Predator Nullcline, H=0.8, D=1 In this case, there is no intercept with x- axis. By increasing asymptotic behavior as. we lost it. We can observe The prey nullcline has a horizontal asymptote at 14

3) Figure 8: Predator Nullcline, H=0.8, D=1.3 If we increase further, the nullcline bends up. This nullcline has absolutely different behavior then predator nullclline in the Rosenzweig- MacArthur model. For small values of it changes. it is almost a vertical straight line but with increasing 4.2 Prey nullcline ( ) By solving the equation 15

for was obtained This nullcline is too complicated to be analyzed. It is not monotonically increasing or decreasing and is not just concave down or concave up. The behavior is more complicated. This nullcline depends only on the two parameters and. x-intercept This curve intercepts the x-axis at. y-intercept This curve intercepts the y-axis at. The following behavior can be observed when and varies. 16

K=0.5 K=0.8 K=1 K=1.5 K=2.5 K=5 Figure 9: Behavior of the prey nullcline for different values of K This nullcline is obviously not parabolic any more. 4.3. Coexistence equilibria We plotted the nullclines for and fixed and being varied. We observed several situations and saw that there are two dividing cases that determine number of coexistence equilibria. Case I is when two non-zero nullclines have a first tangent contact at one point. This is the situation when there is a creation of coexistence equilibria that splits then into 2 equilibria. This situation will corresponds to a saddle node bifurcation of coexistence equilibria. Case II occurs when there are two coexistence equilibria and one of them will be lost at. There are two nullcline intercepts and one of them is getting closer to x-axis with increasing and then moves 17

to the 4th quadrant. This corresponds to a transcritical bifurcation. Before this coexistence equilibrium is at the x-axis, the monoculture equilibrium is stable, but once the coexistenceequilibrium crosses the x-axis it becomes unstable. This will lead to a transcritical bifurcation. 4.3. Saddle node bifurcation curve We considered first the situation when two nullclines get closer to each other until they have one common tangent point. We wanted to find a set of parameters and for this case. This situation occurs when the gradient of one nullcline is a non-zero multiple of the gradient of the second one. If we denote non-zero nullclines equations as functions and then the necessary condition is. This condition can be replaced by setting, where is the Jacobian matrix The set of conditions that will yield one contact point saddle node bifurcation is By using Mathematica we got the following results: 18

So the set of all conditions is The solution of this system of 3 equations can be found in Appendix A since it is too complicated to state it inside the paper. We fixed parameter and plotted values of parameter as a function of Here are pictures for different values of On the horizontal axis are values of parameter and on the vertical axis are values of parameter 19

H=0.1 H=0.5 H=0.8 H=1 H=1.5 H=2. Figure 10: Saddle node bifurcation curve On the left side of the curve there are no coexistence equilibrium and on the right side there are 2 of them. Among the line, there is one coexistence equilibrium. 4.4. Transcritical bifurcation A second option was when the nullclines intersect at the x-axis. This is a crutial situation for the monoculture equilibrium because it determines if it is stable or unstable. It can be a transition between 0 and 1 coexistence equilibria, if the nullclines meet for first time at x-axis or between 2 and 1 equilibrium, as is in the case if we lose one equilibrium because it moved into the 4th quadrant. 20

The set of conditions for this situation is Notice that this curve does not depend on values of This curve always passes through the origin. and it has horizontal asymptote at. Figure 11: Transcritical bifurcation (intercept at the x-axis) 21

We combined both pictures of both cases to see how the whole 1st quadrant is divided into regions with different number of equilibria. Figure 12: Saddle Node and Transcritical Bifurcation Curves We found a common point of these two curves, points where nullclines have a first (tangential) contact at the x-axis. This is the situation that divides 0 and 1 coexistence equilibrium. 22

Solution is A common point of these two curves always exists since we assumed. We added this point into our picture. Figure 13: Part of the curve without biological meaning for H=0.5 We observed that the dashed part of the curve corresponds to the situation where there is a tangent contact for negative values of and. Since we are interested in positive values, we looked only at the segment of the curve for. 23

In summary, we show these bifurcation curves for different values of. H=0.2 H=0.5 H=0.8 H=1 H=1.5 H=2.5 Figure 14: Parameter region division Observe that common point is moving to the right side for getting closer to and moving to the left side for getting larger. If, this point is in first quadrant. If, it is located at the origin and if it is in the third quadrant. The parameter space is divided into 3 regions with different numbers of coexistence equilibria. 24

5. Equilibria The number of equilibria are summarized as 0 saddle node 2 transcritical 1 Figure 15: Parameter space picture with number of equilibria for H=0.8 Solving the systems of two equations for and gives equilibria points. 5.1. Extinctiction This is the case when both species have died out. 25

5.2. Monoculture equilibrium This is where only the prey population is alive, while all predators have died out. In order to determined stability of this equilibrium, we computed the determinant of the Jacobian matrix, which relys on the partial derivatives of equations, when and. Let and. Then This value is not new for us because this value of is the same as value of for which an intercept of two nonzero nullclines is at the x-axis. If, so if, the equilibrium is stable and if the equilibrium is unstable. So as long as lies to the right side of the prey nullcline,the monoculture equilibrium is stable. When the prey nullcline 26

crosses, we either create one stable coexistence equilibrium or loose one of two coexistence equilibria and monoculture equilibrium looses it stability at a transcritical bifurcation 5.3. Coexistence equilibria There appeared to be at most two coexistence equilibria.they depend on parameters. and Since these equilibria are difficult to analyze, we used Mathematica to show what behaviors and which bifurcations is possible to observe for the system (7). Without loss of generality we let. For simplicity we set. We will fixed and vary only and. Let us look first at different situations for and. 27

Case I: From Figure 16 it is obvious that we can observe 4 different scenarios. When. 0 saddle node 2 transcritical 1 Figure 16: Set of parameters for H=0.8 1) We chose and consider the phase planes for system (7) for a sequence of increasing K values. 28

K=0.1 K=0.11 K=0.16 K=1 K=2 K=2.75 K=3.3 K=3.345 K=3.45 Figure 17: Behavior of the model for H=0.8, D=0.1 For small values of there is no coexistence equilibrium. There is only monoculture equilibrium that is stable. Then nullclines intercept at one contact point on the x-axis. After increasing the intersection point is moves deeper into the first quadrant and represents a coexistence equilibrium that is stable. The monoculture equilibrium becomes unstable and we can observe a transcritical bifurcation. With even larger we can observe creation of a periodic solution and a lost of stability of the coexistence equilibrium at a Hopf bifurcation. The trajectory spirals out 29

of the equilibrium into a limit cycle. This state appears to hold for all larger values of behavior is typical of the Rosenzweig-MacArthur system.. This 2) K=0.15 K=0.2499 K=0.4 K=1.5 K=2.25 K=2.5 K=2.515 K=2.55 K=2.7 Figure 18: Behavior for H=0.8, D=0.2 We can observe the same behavior for this set of parameters as for. The only difference is that the transcritical and saddle node bifurcation occur at the same value of. 30

3) K=0.2 K=2.5 K=2.96 K=3.3 K=3.5 K=3.7 K=4 K=4.3 K=5 As before we start with no inner coexistence equilibria; there is only a stable monoculture equilibrium. After increasing, nullclines are getting closer and eventually have a contact point inside the first quadrant. This is a creation of a coexistence equilibrium. With larger this equilibrium splits into two points. The upper one is stable (attractor) and the lower one is unstable (saddle). We have observed the saddle node bifurcation. As before, monoculture equilibrium is stable for low values of. When there are two inner equilibria, the monoculture 31

equilibrium is still stable. With increasing, the lower equilibrium approaches the x-axis and then collides with the monoculture equilibrium It then moves to the 4th quadrant and become biologically irrelevant. After losing this second equilibrium, the monoculture equilibrium becomes unstable and we can observe again transcritical bifurcation. The upper equilibrium stays stable for low values of. However, with increasing, we can observe the creation of a periodic solution again by way of a Hopf bifurcation. The periodic solution appears to persist for all larger values of. 4) K=0.1 K=4 K=10 K=21 K=22 K=30 In this case, we can observe a slightly different situation. For small values of, there is again only stable monoculture equilibrium and no coexistence equilibrium. With larger, the prey nullcline approaches the predator nullcline until they have a contact point. Enriching the system even more, this inner equilibirum splits again into two coexistence equilibria but the 32

monoculture equilibria stays stable for all values of K. One equilibrium is a saddle and the other one is a unstable repellor. We do not observe periodic solutions, therefore no Hopf bifurcation occurs. This characteristic of having two unstable coexistence equilibria persists for all large values of. This behavior is not ecologically desirable because the predator will always die out. This means that if is too large (greater than 1), there is no way how to prevent the extinction of predators, and only the prey population will stay alive. Notice how large values of must be in order to observe two coexistence equilibria for, compare to previous cases. Case II: Note that for the model does not support the case where there is only zero or one equilibrium as we saw for smaller values. We looked at three different cases. and 0 saddle node 2 transcritical 1 Figure 19: Three different scenarios for H=1.5 33

1) K=0.4 K=0.4 2 K=0.415 K=0.43 K=1 K=2.97 K=3 K=2.7 K=4 Figure 20: H=1.2, D=0.3 In this enrichment sequence, we can observe the same behavior as in case. There is no coexistence equilibrium for small, only a stable monoculture equilibrium. After nullclines have a contact point inside the first quadrant there is one coexistence equilibrium that splits into two. One of them is a stable attractor and second one is a saddle node. The saddle node will collide with the monoculture equilibrium and switch stability with the monoculture equilibrium, which becomes unstable and at a ranscritical bifurcation. One 34

equilibrium moves into biologically irrelevant 4th quadrant. With increasing a Hopf bifurcation again., we can observe The only difference in behavior for is that we will always observe first the creation of a coexistence equilibrium inside the first quadrant (saddle node bifurcation) and after that a transcritical bifurcation. There is no set of parameters of and that would give us first the creation of a coexistence equilibrium at the x-axis (transcritical bifurcation). 2) K=0.1 K=2 K=2.33 K=2.5 K=2.8 K=3.14 K=3.4 K=4 K=5 Figure 21: H=1.2, D=0.8 35

3) K=0.5 K=5 K=8 K=9.08 K=10 K=15 Figure 22: H=1.2, D=1.2 In this enrichment sequnce a pair of unstable coexistence equilibria are created, but the monoculture equilibrium will always be attracting. Notice that large values of for obtaining a coexistence equilibria when. are necessary Stability region Simulations suggest that the behavior of model solutions for smaller values of depends on the initial conditions for the simulation. We looked at this situatuation more carefuly and the following behavior was observed. 36

Figure 23: Dependence relationship on initial values for H=0.8, D=0.1 and K=3.34 These two pictures are for the same set of parameters and. They differ only in choice of inital conditons. We can see that the behavior depends on the initial condition. For one choice the trajectory spirals towards the attractive equilibrium and for another choice trajectories are spiral out into a limit cycle. This demonstrates that for certain parameter values the model can support "bistability" ( a locally attracting coexistence equilibrium surrounded by an attracting periodic orbit.) The same behavior was observed for different values of parameters. Figure 24: Dependence of behavior on initial conditions for H=0.8, D=0.2, K=2.55 37

Figure 25: H=1.2, D=0.3, K=3 5.4. Hopf Bifurcation We already know values where transcritical bifurcation and saddle node bifurcation occur. We also want to know for which set of parameters we can have a Hopf bifurcation. A Hopf bifurcation occurs when an equilibrium looses its stability at the creation of a periodic solution. In order to have Hopf bifurcation the eigenvalues of the system's Jacobian matrix must be purely imaginary. If is eigenvalue of the Jacobian matrix and its real part is 0 then. In order to find such eigenvalues we want, where is the identity matrix. Let and be equations of non-zero nullclines. The determinant condition becomes 38

This complex number is zero if the real part and the imaginary parts are both equal to zero. Therefore, we have the conditions Since we still work with equilibria we need to have simultaneous solution of our nullclines. The complete set of conditions that must be satisfied in order to have a Hopf bifurcation is Since and 39

we can write this as In our case Simplifications of this system and its solution can be found in Appendix C. If we fix this is a system of 4 equations for 5 unknowns and. We decided to fix also and solve this system for and. Even Mathematica had a hard time to solve this system exactly with the Solve command. It took it about 35 minutes to find a solution which was too complicated to be useful. Therefore, we tried to find a numerical solution and compared it to the exact solutions to check that they were the same. Since the NSolve commmand was the faster way how to solve the system, we choose this approach. A typical calculation took about 2 seconds to get a solution. Once we found a solution we tried different values of and obtained many combinations of values of associated and. Some of them were negative or complex. We focused on positive real ones since they were biologically relevant. After plotting these points it was obvious what shape of curve we might typically expect. We also chose several points and used the Spline commend in Mathematica to obtain a curve where the Hopf bifurcation occurs. This curve was added to Saddle node and transcritical bifurcation curves and the final region divisions for and were obtained. 40

H=0.8 H=5 Figure 26: Hopf bifurcation, single points Parameter space pictures that covers all types of bifucartion are observed. 0 saddle node 2 Homoclinic transcritical 1 Hopf Figure 27: Parameter space region division for H=5 Points coordinates : 41

0 saddle node 2 transcritical 1 Hopf Figure 28: Parameter space region division for H=0.8 Points coordinates: In summary, the behavior of the model is more complicated then behavior of Rosenzweig- MacArthur model. There are several cases that can occur when. Case I: The creation of a coexistence equilibrium at a transcritical bifurcation followed by a Hopf bifurcation. Case II: The creation of a coexistence equilibrium at a saddle node bifurcation and a transcritical bifurcation simultaneously followed by a Hopf bifurcation. Case III: The creation of two coexistence equilibria at a saddle node bifurcation followed by the loss of one coexistence equilibria at a transcritical bifurcation, and a Hopf bifurcation. Case IV: The creation of two coexistence equilibria at a saddle node bifurcation followed by the loss of one coexistence equilibria at a transcritical and a Hopf bifurcation simultaneously. Case V: The creation of two coexistence equilibria at a saddle node bifurcation followed by a Hopf bifurcation and the loss of one coexistence equlibrium at a transcritical bifurcation. Case VI: The creation of two coexistence equilibria at a saddle node bifurcation. 42

5.5. Solution We verified observed behavior by plotting solutions. On the horizontal axis is time and on the vertical axis are the prey density and the predator density. We chose. Figure 29: Solution for H=0.8, D=0.2, K=0.1 For small values of out., monoculture equilibrium is stable and the predator population has died Figure 30: Solution for H=0.8, D=0.2, K=1.5 For larger values, there is a stable coexistence equilibrium. Trajectories spiral in. 43

Figure 31: Solution for H=0.8, D=0.2, K=2.25 For even larger values, oscilliations are getting bigger. Figure 32: Solution for H=0.8, D=0.2, K=2.55 As we stated before, it is possible to observe bistability. For the same parameter values but different initial values, we observe either oscillations getting smaller or oscillations getting smaller which corresponds to spiraling in and spiraling out in the phrase plane. 44

Homoclinic bifurcation We could also observe the case of homoclinic bifurcation. This case was not fully studied in this paper. Here are pictures for values that shows that this situation can happen in our model. K=3.2 K=3.5 K=4.5 K=4.9 K=5.5 Figure 33: Homoclinic bifurcation, H=5, D=0.9. 6. Comparison to Rosenzweig-MacArthur model Nullclines In the Rosenzweig-MacArthur model there are 4 nullclines, the x-axis, the y-axis,a straight line predator nullcline and a parabolic prey nullcline. In our model, there are also 4 nullcllines. The x 45

and y axis are the same, but the others differs significantly. Both of them are more complicated curves with behavior depending on model parameter values. Equilibria In both models there is an extinction equilibrium and monoculture equilibrium. We should not forget that used in our model is a rescaled parameter while in Rosenzweig- MacArthur it is the original not rescaled value. The models differ significantly in their coexistence equilibriua. While in the Rosenzweig-MacArthur model we can observe only one coexistence equilibrium, in our model under certain conditions can observe two coexistence equilibria. Bifurcation In the model developed here it is possible to observe a transcritical bifurcation as in the Rosenzweig-MacArthur model. We can also see a Hopf bifurcation as in the Rosenzweig- MacArthur model. However, the behavior of our model strongly depends on the parameter values for and. Our model shows richer behavior including bistability and Homoclinic bifurcations in contrast to the Rosenzweig-MacArthur model. 7. Further Research Suggestions 1) The influence on different values of values of and could be analyzed. Recall we assumed that and. 2) For which values of parameters and is possible to observe a Homoclinic bifurcation? 3) What is the dependence relationship between initial values and stability of a coexistence equilibrium? For which values of and is it possible to see such a behavior? 46

Table of Figures Figure 1: Three types of functional response... 2 Figure 2: Type I functional response... 2 Figure 3: Type II functional response... 3 Figure 4: Type III functional response... 3 Figure 5: Rosenzweig-MacArthur model - nullclines... 6 Figure 6: Predator nullcline, H=0.8, D=0.5... 14 Figure 7: Predator Nullcline, H=0.8, D=1... 14 Figure 8: Predator Nullcline, H=0.8, D=1.3... 15 Figure 9: Behavior of the prey nullcline for different values of K... 17 Figure 10: Saddle node bifurcation curve... 20 Figure 11: Transcritical bifurcation (intercept at the x-axis)... 21 Figure 12: Saddle Node and Transcritical Bifurcation Curves... 22 Figure 13: Part of the curve without biological meaning for H=0.5... 23 Figure 14: Parameter region division... 24 Figure 15: Parameter space picture with number of equilibria for H=0.8... 25 Figure 16: Set of parameters for H=0.8... 28 Figure 17: Behavior of the model for H=0.8, D=0.1... 29 Figure 18: Behavior for H=0.8, D=0.2... 30 Figure 19: Three different scenarios for H=1.5... 33 Figure 20: H=1.2, D=0.3... 34 Figure 21: H=1.2, D=0.8... 35 Figure 22: H=1.2, D=1.2... 36 Figure 23: Dependence relationship on initial values for H=0.8, D=0.1 and K=3.34... 37 Figure 24: Dependence of behavior on initial conditions for H=0.8, D=0.2, K=2.55... 37 Figure 25: H=1.2, D=0.3, K=3... 38 Figure 26: Hopf bifurcation, single points... 41 Figure 27: Parameter space region division for H=5... 41 Figure 28: Parameter space region division for H=0.8... 42 Figure 29: Solution for H=0.8, D=0.2, K=0.1... 43 Figure 30: Solution for H=0.8, D=0.2, K=1.5... 43 Figure 31: Solution for H=0.8, D=0.2, K=2.25... 44 Figure 32: Solution for H=0.8, D=0.2, K=2.55... 44 Figure 33: Homoclinic bifurcation, H=5, D=0.9.... 45 47

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