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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: h? = -Mh + Ahp p? = Gp - Bhp where M, A, G, B are positive constants. (a) Find both equilibrium points h?, p?. (b) Linearize the differential equation about the nonzero equilibrium h?, p?, and put the linearized model into state-space form x? = Ax where x = [?h, ?p]?. (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.

          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:

h? = -Mh + Ahp
p? = Gp - Bhp

where M, A, G, B are positive constants.

(a) Find both equilibrium points h?, p?.
(b) Linearize the differential equation about the nonzero equilibrium h?, p?, and put the linearized model into state-space form x? = Ax where x = [?h, ?p]?.
(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.

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:

h? = -Mh + Ahp
p? = Gp - Bhp

where M, A, G, B are positive constants.

(a) Find both equilibrium points h?, p?.
(b) Linearize the differential equation about the nonzero equilibrium h?, p?, and put the linearized model into state-space form x? = Ax where x = [?h, ?p]?.
(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.

Added by Ryan M.

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