How many time steps does it take for iterative model to reach equilibrium?

templatedynamic

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I am not a math student, merely a biology student trying my hand at behavioral modeling, so I apologize in advance if the terminology or notation I use isn't appropriate for the field.

Individual ants within a colony of N ants can be in one of two states: it can perform an action or it does not. The decision to start a task is based on the current need for the task, S (which can vary between 0 and 1). The higher S, the more likely an ant will start performing an action. Every timestep, S increases by the constant C and decreases every time an ant performs the task. The probability that an ant will start a task is S2/(S2+.25). The probability that an ant will continue doing a task given that it is already doing it is D. The current value of S is given by equation 1:

(1) Si+1 = Si + C - (2C/N * ni)

If t timesteps pass, then equation 2 gives the value of S:

(2) St= Si + (C * t) - (2C/N * ni) - (2C/N * ni+1) - (2C/N * ni+2) - ... (2C/N * nt)

The number of active ants updates with equation 3:

(3) ni+1 = (N-ni) * (Si2/(Si2+.25)) + (ni * D)

When D = .5, C = .1, I know that eventually this system of equations will eventually converge on the point S = .5, n = N/2 (see attached figure of vector field: the vertical line is the point at which S stops changing, and the curve is the line at which n stops changing, thus their intersection point is the equilibrium point). My question is for a given Sinitial and ninitial set of values, at what value of t will the system converge at this point?
 

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The existence of a fixed point does not entail that it is attractive.
 
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