Two interacting species with densities X.Y are modelled by the equations dX X(a - bx - cY) (1) dt dY = Y(-d+ex), (2) dt

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Two Interacting Species With Densities X Y Are Modelled By The Equations Dx X A Bx Cy 1 Dt Dy Y D Ex 2 Dt 1 (40.71 KiB) Viewed 36 times

Two interacting species with densities X.Y are modelled by the equations dX X(a - bx - cY) (1) dt dY = Y(-d+ex), (2) dt where a, b, c, d, and e are positive real constants. (a) Briefly discuss the model, identifying carefully the types of species-species interactions involved. What is the carrying capacity of the species with density X? (Hint: For the carrying capacity, consider the differential equation for X when Y = 0.1 (b) Re-scale the equations, using the new variables and parameters: bx cY T=at, d db ae (c) Assuming that < 1, find all steady states of the re-scaled system of equations that you found in part (6). Determine whether each steady state is locally stable or unstable. (d) Carefully sketch the phase plane for the system that you found in part (b), again assuming that < 1. (e) Assuming that a = d, give conditions on the parameters of the original system (equations (1) and (2)) for the interior steady state (i) to be biologically meaningful and (ii) to also be a stable node. (f) For the case a = b = d = 1 and e = 2, briefly sketch the phase plane of the re-scaled system from part (b). indicating the behaviour in the vicinity of the interior steady state. a