THETA-LOGISTIC PREDATOR PREY

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1 THETA-LOGISTIC PREDATOR PREY What are the assumptions of this model? 1.) Functional responses are non-linear. Functional response refers to a change in the rate of exploitation of prey by an individual predator resulting from a change in prey density (Stiling, 1998). 2.) Prey population growth is density dependent. When we say density dependent, this means that the influence on individuals does not vary with the number of individuals per unit area in the population (Stiling, 1998). It can be seen that lesser assumptions were made as compared to the Lotka-Volterra model. This makes the Theta-Logistic model more realistic ( Dynamics of Prey dn = rn[1- (N/K) θ ] f(n)p dt where r- rate of increase for prey population N- prey population size rn[1- (N/K) θ ] - prey population growth rate in the absence of predator - highlighted equation is algebraically equivalent to (K-N)/K of the Lotka- Volterra equation except that a density dependence term (exponential θ) is added θ density dependence term; if this variable is large, density dependence is strong only when prey population size is near the carrying capacity, K. On the other hand, small values for this variable would show that density dependence is strong at low population density f(n) functional response of predator P predator population size Dynamics of Predator dp = sp[f(n) D] dt where s- efficiency at turning food into offspring P- predator population size sp population size at constant times D- number of prey needed for predator to replace itself in the next generation f(n) functional response Note that sp is an exponential growth curve damped by f(n) and D needed before individual predators contribute to population growth.

2 Two implicit assumptions are seen in this equation: 1.) Predator population density does not effect an individual predator s chances of birth and death directly. 2.) The number of surviving offspring produced by a predator is directly proportional to the amount of prey it consumes The Functional Response One of the assumptions given for the Theta-Logistic model is that functional response is non-linear. There are 3 types of functional response in this model. Type 1: f = C 1 N 2: f = C 2 N/(1 + h 2 C 2 N) 3: f = C 3 N 2 / (1 + h 3 C 3 N 2 ) where C constant N number of prey h handling time Type 1 f = C 1 N C 1 N capture rate Response is linear where a constant C is multiplied by population density N Handling time is absent Figure 1. Graphical representation of Type I functional response Type 2 f = C 2 N/(1 + h 2 C 2 N) C 2 N encounter time when not handling Asymptotes at 1/h (maximum rate where prey can be captured), h is handling time Accounts for satiation and handling time Figure 2. Graphical representation of Type 2 functional response

3 Type 3 f = C 3 N 2 / (1 + h 3 C 3 N 2 ) C 3 N 2 capture rate when not handling Asymptotes at 1/h (maximum rate where prey can be captured), h is the handling time Accounts for satiation, handling time and prey switching Figure 3. Graphical representation of Type 3 functional response HOW TO RUN THETA LOGISTICS MODEL ON POPULUS Open Populus 5.1 for Windows. This program can be downloaded from Under the Model tab, choose Continuous Predator-Prey Models. A new window will open. Choose -Logistic for model type. For termination conditions, choose run until steady state. Tick P vs. N for graph type. Input values for predator and prey where everything is equal to 1 except for g (equivalent to sp) where g = 0.1. To see the graph, click the View tab and a new window will open showing the graph. SIMULATIONS The values of the variables were kept constant to see the effects of increasing or decreasing a particular variable. Manipulations and suggested values of variables used were obtained from the worldwide web ( One should keep in mind that the program Populus can only accept up to certain values. The values are listed below: P and N:0-100,000 r and D: 0-5 C: 0-999

4 Type 1 Functional Response Effect of increasing the values of carrying capacity, K Figure 4. K=0.5 Figure 5. K=1 Figure 6. K=5 When the carrying capacity is lower, the slope of the prey density is steeper as compared to a higher value of K. This gives us an idea that the density dependent term has a lesser effect for a higher value of K since results had shown that the model is similar as that of Lotka- Volterra. It should be recalled that one of the assumptions of the Lotka-Volterra model is that the prey lives in a density-independent environment. Effect of increasing rate of increase of prey population, r Figure 7. r = 0.5 Figure 8. r=1

5 Type 2 Functional Response Comparison with Type 1 Functional Response Figure 9. r =3 It can be seen that by increasing the rate of increase of prey population while keeping the carrying capacity constant also increases the prey population. For a linear functional response, predators should benefit from increase in prey population. Figures 10 & 11. Comparison of Type 1 (left) and Type 2 (right) functional response The type 2 response does not show a linear graph as that of type 1. It can also be observed that greater number of predators are observed in the type 2 response. This gives us an idea that for a type 2 predator functional response, larger number of predators are needed to reduce the population size of the prey.

6 Effect of doubling handling time, h Figures 12 & 13. Effect of doubling handling time, h. (left) h=1. (right) h=2. It can observed that the graph of prey population where handling time was doubled has a more stable curve as compared to the first graph. It should be recalled that 1/h is the asymptote or maximum rate of which prey can be captured. Doubling the handling time can decrease the rate from 1 to 0.5 of prey captured by predator per time period. This would then lead to more stable cycles ( Effect of increasing density dependence term, θ Figures 14, 15, 16 & 17. Effect of decreasing values of θ. (Counterclockwise from top left) Decreasing values of density dependence.

7 Longer prey density graphs are observed for higher values of θ. This implies that the prey population grows for a longer time until density dependent term slows the change of prey population per change in time. Effect of changing capture rate, C Figures 18, 19 & 20. Effect of increasing capture rate C. (Clockwise l-r) C=1, C=10 and C= 100. The predator density shifts downward as capture rates increases. This gives use an idea that lesser prey are needed for the predator to maintain its growth. POSTSCRIPT One should take note that data can be further manipulated. It is up to the reader to explore more on the -Logistic model in Populus. This program can be downloaded from REFERENCES Populus version 5.1 for Windows. Downloaded September, Stiling, Peter Ecology: Theories and Applications 2 nd ed. Prentice Hall International. Accessed October 9, 2002.