templatedynamic
New member
- Joined
- Dec 19, 2017
- Messages
- 1
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?
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?
Attachments
Last edited: