Exercise #2 Below is the basic Lotka-Volterra predation model (not competition and does not incorporate density dependence) shown graphically as the isoclines of predator and prey. (a) Define "isocline" as it relates to predator-prey phase diagrams (b) Label the axes and label the legend (which line represents predator isocline and which line represents prey isocline) (c) Identify the joint equilibrium. (d) Show with the use of arrows and vectors the joint movement of predator and prey populations for each of the regions of the graphs (each region is defined by a number, with reference to the isoclines). (e) Indicate which triangle—each representing a different starting point for the system—would result in lower amplitude cycles in the stable limit cycle. 2 1 r/? 3 4 y-axis title: x-axis title: q/? (d) Why do arrows point the way they do? Region 1 Region 2 Region 3 Region 4
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In other words, it is a line where the growth rate of either predator or prey population is the same at all points on the line. Show more…
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Below is the basic Lotka-Volterra predation model (not competition and does not incorporate density dependence) shown graphically as the isoclines of predator and prey: (a) Define "isocline" as it relates to predator-prey phase diagrams. (b) Label the axes and label the legend (which line represents predator isocline and which line represents prey isocline). (c) Identify the joint equilibrium. (d) Show with the use of arrows and vectors the joint movement of predator and prey populations for each of the regions of the graphs (each region is defined by a number; with reference to the isoclines). (e) Indicate which triangle - each representing a different starting point for the system - would result in lower amplitude cycles in the stable limit cycle. (d) Why do arrows point the way they do? Region 1 Region 2 Region 3 r/α Region 4 q/β X-axis title:
Adi S.
5. The Lotka-Volterra model is often used to describe the interactions between two species (predator and prey) in an ecosystem. The number (or concentration) of predators (hunters, e.g. foxes) is h, and the number (concentration) of prey (e.g. rabbits) is p. The predators die off at a rate M due to old age, but increase at a rate proportional to the number of prey available to eat. The prey multiply at a rate G (assuming sufficient food supply) but are eaten at a rate proportional to the number of predators. The (nonlinear) model predicts that the populations will change over time according to the following equation: ḣ = -Mh + Ahp ṗ = Gp - Bhp where M, A, G, B are positive constants. (a) Find both equilibrium points h₀, p₀. (b) Linearize the differential equation about the nonzero equilibrium h₀, p₀, and put the linearized model into state-space form ẋ = Ax where x = [͈h, ͈p]ᄄ. (c) Let the output be the number of rabbits in the system, and consider an input as the number of foxes (suppose you can add or subtract foxes). Write the input and output vectors in their linearized forms. (d) What are the poles of the system (eigenvalues of the A matrix)? (e) Is the linearized system stable? Describe its behavior.
Suman K.
Predator-Prey Model. The Volterra-Lotka predator-prey model predicts some rather interesting behavior that is evident in certain biological systems. For example, suppose you fix the initial population of prey but increase the initial population of predators. Then the population cycle for the prey becomes more severe in the sense that there is a long period of time with a reduced population of prey followed by a short period when the population of prey is very large. To demonstrate this behavior, use the vectorized Runge-Kutta algorithm for systems with h = 0.5 to approximate the populations of prey x and of predators y over the period [0, 5] that satisfy the Volterra-Lotka system x' = x(3 - y), y' = y(x - 3) under each of the following initial conditions: (a) x(0) = 2, y(0) = 4. (b) x(0) = 2, y(0) = 5. (c) x(0) = 2, y(0) = 7.
Sri K.
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