Exercise 3. This problem is a modeling discussion around one parameter. The model below represents a predator-prey model. Let’s say R represents rabbits and F are the foxes. The way we model the equations we see that the rabbits, in the absence of foxes, follow a logistic equation and stabilize at a population of 100. On the other hand, without rabbits, the foxes follow a decaying exponential and go extinct to model the fact that they are dependent on the prey for their survival. The predation terms −5RF and 2RF have obviously similar forms as in an epidemic but they are not equal since the first represent predator efficiency (the number of rabbits killed) and the sec- ond represents food intake (the growth due to feeding in the fox population). α represents a harvesting (hunting) effect. We assume that the harvesting happens at the SAME rate α. R′ =(100−R)R−5RF −αR, F ′ = 2RF − 20F − αF. (a) Find the NON-ZERO equilibrium point (the one in which both R and F are non-zero, also known in ecology as the co-existence equilibrium). This will be, of course, in terms of α (b) Find the Jacobian and establish a condition on α such that this equilib- rium is stable (c) Which species is more affected by harvesting? Could you explain why the two species are NOT equally affected although the harvesting happens at the same rate? (d) Attach three PPLANE figures to illustrate the following cases: co-existence without harvesting (α = 0), co-existence with harvesting (α = 20), fox pop- ulation extinction (α = 80)
Exercise 3. This problem is a modeling discussion around one parameter. The model below represents a predator-prey model. Let’s say R represents rabbits and F are the foxes. The way we model the equations we see that the rabbits, in the absence of foxes, follow a logistic equation and stabilize at a population of 100. On the other hand, without rabbits, the foxes follow a decaying exponential and go extinct to model the fact that they are dependent on the prey for their survival. The predation terms −5RF and 2RF have obviously similar forms as in an epidemic but they are not equal since the first represent predator efficiency (the number of rabbits killed) and the sec- ond represents food intake (the growth due to feeding in the fox population). α represents a harvesting (hunting) effect. We assume that the harvesting happens at the SAME rate α.
R′ =(100−R)R−5RF −αR, F ′ = 2RF − 20F − αF.
(a) Find the NON-ZERO equilibrium point (the one in which both R and F are non-zero, also known in ecology as the co-existence equilibrium). This will be, of course, in terms of α
(b) Find the Jacobian and establish a condition on α such that this equilib- rium is stable
(c) Which species is more affected by harvesting? Could you explain why the two species are NOT equally affected although the harvesting happens at the same rate?
(d) Attach three PPLANE figures to illustrate the following cases: co-existence without harvesting (α = 0), co-existence with harvesting (α = 20), fox pop- ulation extinction (α = 80)
Step by step
Solved in 3 steps with 2 images
