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2

I think your issue is mainly with the interpretation of the isoclines. If you assume this standard predator-prey model: $$\frac{dN_1}{dt} = rN_1-pN_1N_2$$ $$\frac{dN_2}{dt} = apN_1N_2-mN_2$$ you get the isoclines: $$N_2 = \frac{r}{p}$$ $$N_1 = \frac{m}{ap}$$ with the first isocline referring to prey and the second to predators. These isoclines mean ...

-1

I am going to try and cover the math with a few more words, to show that your intuition is in fact, correct (which with math, is not always given). In the differential equation you wrote $$\frac{d N_1}{dt} = r_1 N_1 - p_1 N_1N_2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)$$ the numbers $r_1$ and $p_1$ are simply constants that give a ...

1

You can write these equations in the form of ODEs but it is not really essential. This is my analysis (just the model formulation and not literature): The effects of interactor species may or may not be dependent on the population of the species that is being analysed (recipient). Now you have to model based on how you would realistically expect the ...

2

This is probably not a very satisfactory answer. The very next step in Murray's book (3rd Ed) there is a non-dimensionalised version of the model (eq 3.32) which looks like the one given in Boyce's book (10th Ed) except $\epsilon_i$ and $\sigma_i$ (eq 2, Ch 9). If you plug in $\epsilon_i = \sigma_i = 1$ in Boyce's book you get the same fixed point expression ...

1

What you are looking for can easily found in any undergrad text on ecology and it goes by the name (competitive) Lotka-Volterra Equations. This is a system Ordinary Differential Equations and in my opinion, can easily be converted to recursive or difference equations. Your guess with logistic growth is fairly close to these equations. You can remove the term ...

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