# Replicator dynamics giving probabilities greater than 1?

I have a linear replicator equation and I want to simulate its dynamics to find the resting point and hopefully the Nash equilibrium.

I defined strategies' payoff as $\vec f = C*\vec p+ \vec b$, where $C$ is a community matrix similar to the one in Lotka volterra models, $p$ is the vector of probabilities for each strategy and b is a vector of constants. From what I understand, this is very similar to the replicator dynamics for a linear model, except that I'm adding a constant.

However, when I iterate over $\vec p_{t+1} = \vec p_t+ \vec p_t (\vec f- \sum\vec f \vec p_t)$, $\vec p_{t+1}$ frequently gets values greater than one or smaller than zero, which from my understanding is not suppose to happen.

An example to show the issue:

$C = \begin{pmatrix} -1 && -1.1 && 0 \\ 1 && -1.0 && -1.1 \\ 0 && 1 && 0 \end{pmatrix}; \space \vec b= \begin{pmatrix} 0.7 \\ 0 \\ 0 \end{pmatrix}; \vec p_0 = \begin{pmatrix} 0.8 \\ 0.9 \\ 0.25 \end{pmatrix}$

In this situation, the second element of $\vec p_1$ equals 1.448550!

What could be causing this error? Am I calculating the replicator dynamics wrong? Does adding a constant to fitness "breaks" the replicator dynamics? Is my understanding wrong and replicator dynamics can go above 1 or below 0?

• Can you please describe the meaning behind your equation because it does not really look like Lotka-Volterra equations? Lotka-Volterra equations are explained in this post. – Remi.b Aug 23 '17 at 14:27
• Sorry, I mean that payoff of a strategy depends on the frequency of use of these strategies. This is represented by the matrix C. In the example, the -1 on the top left indicates that the more strategy 1 is used, the smaller is it's payoff. Likewise the more strategy 2 is used the smaller is the payoff of strategy 1 by a factor of -1.1. however irrespective of strategies distribution in the population there is a baseline payoff b for every strategy. – JMenezes Aug 23 '17 at 14:39
• It is hard to tell what you are trying to compute. It is not obvious to me looking at the equation that $\arrow p$ must contain probabilities. Also, it is a bit weird that you are adding a baseline at every generation. Of course, the baseline fitness / probability (or whatever is represented of $\arrow p$) will very quickly become greater than 1. – Remi.b Aug 23 '17 at 14:46
• $\vec p$ is supposed to represent the proportion of time exercising a behavior (eg feeding). Every individual, ie every row of C and every element of in p, is supposed to have a value going from 0 to 1, representing how much of the time it spends doing said behavior. but the benefit of foraging more or less depends on how much other individuals are doing. Thus I want to know if there is any equilibrium point where all individuals stick with a certain proportion of time feeding. Hope it helps, and thanks for the time invested – JMenezes Aug 23 '17 at 15:02
• If I'm to provide an answer to this, I first need to know: How did you obtain the values for the $C$ matrix? How did you come up with the definition of $f$? What purpose does the $b$ vector serve? And lastly, in your recurrence relation (definition of $p_{t+1}$), regarding the summand expression, are you multiplying the $f$ vector with the $p$ vector? I tried looking at your mathjax but it still wasn't clear from where you didn't bracket anything. Also, at first glance, shouldn't it be $p_{t+1} = p_{t} * p_{t}(...)$? i.e., multiplication between the two terms, and not addition? – user22020 Aug 23 '17 at 16:38