3) (2 points) Consumer-resource dynamics: Consider a system with the following parameters: - Resource growth rate r = 0.5 - Consumer attack rate a = 0.1 - Consumer conversion efficiency e = 0.2 - Consumer maintenance cost m = 0.3 Draw the isoclines for this system on the following state-space graphs. 1. Please ensure you label all axes 2. If you do any calculations, show your work. 3. Add growth vectors (arrows) to parts of your state-space depending on the placement of isoclines Resource isocline Combined isoclines Consumer isocline
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This occurs when the rate of resource growth equals the rate of resource consumption. The rate of resource growth is given by rR, where R is the resource population. The rate of resource consumption is given by aRC, where C is the consumer population. Setting Show more…
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Adi S.
Question 3. You are studying the predator-prey relationship between lizards (predator) and beetles (prey). The intrinsic growth rate of beetles in the absence of lizards (r) is 0.2 per week, and the mortality rate of lizards in the absence of beetles (m) is 0.1 per week. The attack rate (a) is 0.004, and the efficiency at which beetle biomass is converted into lizard biomass (b) is 0.25. Assuming the interaction follows Lotka–Volterra dynamics, changes in prey and predator population sizes are determined by the following equations: dNprey/dt = rNprey - aNpreyNpred dNpred/dt = b(aNpreyNpred) - mNpred A. If there are 35 lizards and 160 beetles in the two populations, approximately how many beetles will be killed per week? Show your work. B. Complete the above phase plane by adding the zero growth isoclines for each species, such that dNprey/dt = 0 when Npred = r/a and dNpred/dt = 0 when Nprey = m/ba. Label the axes and lines; show your work. C. Given initial population sizes of Nprey = 160 beetles and Npredator = 35 lizards, what are the expected short-term (one time step) dynamics for each population (i.e., do they increase, decrease, or stay the same)? Explain. D. What are the expected long-term dynamics for the two populations described above (i.e., do they go extinct, reach carrying capacity, or coexist in some way)? Be specific. E. In the Lotka–Volterra predator–prey model, an increase in the attack rate (a) should have what effect on the long-term average number of predators and prey? Explain.
Suman K.
The Rosenzweig-MacArthur model is a consumer- resource model similar to that from Exercise $30,$ but with a different consumption function. A simplified version is $$R^{\prime}=R(K-R)-\frac{R}{a+R} C \quad C^{\prime}=\frac{R}{a+R} C-b C$$ Suppose that all constants are positive and that $K>a b /(1-b)>0 .$ Construct the phase plane, including all nullclines, equilibria, and arrows indicating the direction of movement in the plane.
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