## Mathematical Biology (Modelling)

Consider the two species competition model given by
da/dt = [λ1 a /(a+K1)] - r_(ab) ab - da, (1)
db/dt = [λ2 b *(1-b/K2)] - r_(ba) ab , t>0, (2)
for two interacting species denoted a=a(t) and b=b(t), with initial conditions a=a0 and b=b0 at t=0. Here λ1, λ2, K1,K2, r_(ab), r_(ba) and d are all positive parameters.
(a) Describe the biological meaning of each term in the two equations.

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A series expansion of 1/(a+K1), gives
1/(a+K1) ≈ (K1 -a)/ K1 ^2 + O (a^2)
Now,
da/dt = [λ1 a * (a+K1)/ K1^2] - r_(ab) ab - da,

λ1 a represents the exponential growth of population
da represents the exponential decay of population
λ1 is the growth rate
d is the decay rate
r_(ab) ab is an interaction term of a and b
r_(ab) can be thought of as the decrease in growth rate of species "a" due to the presence of species "b".
The first term of RHS equation 1: [λ1 a * (a+K1)/ K1^2] represents logistic growth at a rate λ1 with carrying capacity K1.

db/dt = [λ2 b *(1-b/K2)] - r_(ba) ab ,
λ2 b represents the exponential growth of population
λ2 is the growth rate
r_(ba) ab is an interaction term of a and b

r_(ba) can be thought of as the decrease in growth rate of species "b" due to the presence of species "a".
The first term of RHS equation 2: [λ2 b *(1-b/K2)] represents logistic growth at a rate λ2 with carrying capacity K2.