How Do Interactions Between Two Species Model Their Population Dynamics?

[ra_forever8]

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.

=>
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

 

[Ackbach]

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.

 

[ra_forever8]

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?

 

[Ackbach]

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.

 

[CellCoree]

Your answer is correct. In addition, r_(ab) and r_(ba) represent the interaction between the two species, where r_(ab) is the effect of species b on species a and r_(ba) is the effect of species a on species b. These interactions can either be positive (facilitation/competition) or negative (competition/predation), depending on the sign of the parameters.