I was wondering if someone could help me form a differential equation from the following game:
A population consists of two types of diets, fish and veg. People play a with every other person and the game has the following payoff matrix, indicating the better diet:
| Fish | Veg
------------------------------
Fish | (0,0) | (1, 1-2c)
Veg | (1-2c, 1) | (1-c, 1-c)
At the end of a round a person compares to another random person. If the other person's total payoff, $\pi_j$, is larger than the first person's, $\pi_i$, the first person copies the diet of the other person with a probability proportional to differences in performance, $\pi_j - \pi_i$
This is how I think I need to approach it, but I'm new to this and it's all confusing me a little!
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Let $x$ denote the proportion of fish diets (and $1-x$ denote the proportion of veg diets) and let $\pi$ denote the payoffs per time step:
$\pi_f = (1-x)(1)\\ \pi_d = x(1-2c) + (1-x)(1-c)$
I think change in proportion of fish diets is this, but I'm not sure:
change in fish = -(fish converted to veg) + (veg converted to fish) $\frac{\delta x}{\delta t} = -x(1-x)(\pi_f-\pi_v) + x(1-x)(\pi_v+\pi_d)$
If someone could let me know whether I'm doing this correctly and how best to approach this then I would be very grateful.
Thanks