A population biologist uses the following variants of the Lotka-Volterra model t

Question

A population biologist uses the following variants of the Lotka-Volterra model to model two different ecosystems. each with two populations of animals, x(t) and y(f), where x(t) is the population of animal T at time t, and y(t) is the population of animal y at time t. For each of the two variants, decide if x and y cooperate with each other [increasing x also increases y's growth-rate and vice-versa); x and y compete with each other [increasing x decreases y's growth rate and vice-versa); or one of the animals parasitizes the other [increasing x increases y's growth-rate, but increasing y decreases x's growth rate, or vice-versa).

Explanation / Answer

(a)

x' = 0.12x - 0.0006x^2 + 0.00001xy

y' = 0.08x + 0.00004xy

Let , x = y = 1

x' = 0.12 - 0.0006 + 0.00001 = 0.12059

y' = 0.08 + 0.00004 = 0.08004

Let,

x = y = 2

x' = 0.12 - 0.0006*4 + 0.00001*4 = 0.09604

y' = 0.08*2 + 0.00004*4 = 0.16016

So, increasing x increases y but increasing y decreases x So the condition is one of the animal parasitizes the other.

(b)

x' = 0.15x - 0.0002x^2 - 0.0006xy

y' = 0.2y - 0.00008y^2 - 0.0002xy

Let, x = y = 1

x' = 0.15 - 0.0002 - 0.0006 = 0.1492

y' = 0.2 - 0.00008 - 0.0002 = 0.19972

Let, x = y = 2

x' = 0.15 - 0.0002*4 - 0.0006*4 = 0.1468

y' = 0.2*2 - 0.00008*4 - 0.0002*4 = 0.39888

So, increasing x increases y but increasing y decreases x So the condition is one of the animal parasitizes the other.

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