How Do Interactions Between Two Species Model Their Population Dynamics?

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The discussion focuses on a two-species competition model represented by differential equations that describe the population dynamics of species a and b. Key terms include λ1 and λ2 for growth rates, K1 and K2 as carrying capacities, and r_(ab) and r_(ba) as interaction terms that negatively impact both populations. The equations illustrate logistic growth and the effects of competition, where increased populations of one species reduce the growth rates of the other. The interaction terms r_(ab) and r_(ba) signify how the presence of one species affects the growth of the other. Overall, the model emphasizes the complexities of species interactions in ecological dynamics.
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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
what does r_(ab) ab represent?
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
what does r_(ba) ab represent?
The first term of RHS equation 2: [λ2 b *(1-b/K2)] represents logistic growth at a rate λ2 with carrying capacity K2.

Kindly please check my answer. thank you
 
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I would agree that the first two terms are logistic growth, as in your other thread. $da$ is a decay rate for population $a$. The $r_{ab}ab$ is an interaction term. As either population increases, this term starts to affect both populations negatively. Now, if $da/dt$ had $+r_{ab}ab$ and $db/dt$ had $-r_{ab}ab$, then you'd have a predator-prey model. In this case, both populations suffer when there are interactions; this is consistent with the idea of competing species.
 
is r_(ab) ab an interaction term of a and b?
is r_(ba) ab 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".
r_(ba) can be thought of as the decrease in growth rate of species "b" due to the presence of species "a".

did I define r_(ab) and r_(ba) correctly?
 
grandy said:
is r_(ab) ab an interaction term of a and b?
is r_(ba) ab 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".
r_(ba) can be thought of as the decrease in growth rate of species "b" due to the presence of species "a".

did I define r_(ab) and r_(ba) correctly?


Yes, that looks good to me. By the way, I would recommend a more uniform font when you're writing online. It makes things easier to read.
 
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