dS/dt = f(S) = kS(1-S/N)(S/M-1)
of a fox squirrel population from the previous section. Suppose that the parameters
M and k remain relatively constant over the long term but as more people move into
the area. the parameter N (the carrying capacity) decreases.
(a) Assuming that M <= N, sketch the graph of the function f(S) for fixed values
of k and M and several values of N.
(b) At what value of N does a bifurcation occur?
(c) How does the population of fox squirrels behave if the parameter N slowly and
continuously decreases toward the bifurcation value?
what's the trouble? Expand the right side and you have
dS/dt = k/MN (-S^3 + (M+N)S^2 - MNS)
That's just a polynomial, which you can easily integrate to find S(t)
for purposes of integration n k m are constant
then
ds/dt = -ks (s^2/nm - S/n -s/m +1)
= -k [ s^3/nm - s^2/n - s^2/m + s]
=k[ -s^4/4nm +s^3/3n + s^3/3m -s^2/2 + c]
You can play with this perhaps
http://www.wolframalpha.com/input/?lk=3&i=plot++-s^4%2F4n+%2Bs^3%2F3n+%2B+s^3%2F3+-s^2%2F2
I ran it for m = 1
you can try it with various values of n
(a) To sketch the graph of the function f(S) = kS(1-S/N)(S/M-1) for fixed values of k and M and several values of N, we can follow these steps:
1. Choose fixed values for k and M. Let's say k = 1 and M = 100 for simplicity.
2. Select several values for N. Let's consider N = 50, N = 75, and N = 100.
Now, we can proceed to plot the graph.
For N = 50:
- When S = 0, f(S) = 0.
- As S increases, f(S) initially increases, reaches a maximum, and then decreases.
- As S approaches N, f(S) approaches zero.
- Beyond N, f(S) becomes negative.
- The graph's shape resembles an upside-down "U" with its vertex at the maximum point.
For N = 75:
- The graph will follow a similar pattern to the one described for N = 50, but the maximum value will occur at a higher S value.
- The shape of the graph will still resemble an upside-down "U" but flatter compared to the N = 50 case.
For N = 100:
- When S = 0, f(S) = 0.
- As S increases, f(S) initially increases, reaches a maximum, and then decreases.
- But this time, the maximum value occurs at S = M since N = M.
- Beyond N, f(S) becomes negative.
- The graph's shape resembles a symmetric "S" with its maximum point at S = M.
(b) A bifurcation occurs when the shape of the graph changes abruptly, leading to multiple steady-state solutions or different population behaviors. In this case, a bifurcation occurs at N = M.
(c) If the parameter N slowly and continuously decreases toward the bifurcation value, the population of fox squirrels behaves as follows:
- Initially, the squirrel population exhibits stable and predictable growth patterns, following the shape of the graph as described in part (a).
- As N gradually approaches the bifurcation value N = M, the population behavior becomes more complex and unpredictable.
- The squirrel population may exhibit sudden changes in growth patterns, multiple steady-state solutions, or even chaotic behavior.
- It becomes challenging to determine the long-term behavior of the population, and small changes in N can result in significant changes in the population dynamics.
- Therefore, as N approaches the bifurcation value, it is difficult to predict or control the population behavior accurately.
(a) To sketch the graph of the function f(S) = kS(1-S/N)(S/M-1), we need to choose specific values for k, M, and N. Let's assume k = 1 and M = 10 for simplicity, and plot the function for several values of N.
Start by selecting a range of values for S, let's say from 0 to 20. Then, for each value of N, calculate f(S) by substituting the values of S, k, M, and N into the function.
For example, if N = 10, the equation becomes f(S) = S(1 - S/10)(S/10 - 1). Calculate the corresponding values of f(S) for each value of S within the chosen range, and plot the points on a graph.
Repeat this process for different values of N, such as N = 15, N = 20, and so on. Connect the points to create a smooth curve for each value of N.
(b) A bifurcation occurs when there is a sudden qualitative change in the behavior of the system as one or more parameters cross a critical threshold value. In this case, as N decreases, there is a specific value of N where a bifurcation occurs.
To determine the value of N at which the bifurcation occurs, examine the behavior of the function f(S) as N decreases. Look for changes in the number and stability of equilibrium points or any sudden shifts in population behavior.
For example, when N is large, the function may have a stable equilibrium point where the population size stabilizes around. But as N decreases and approaches the bifurcation point, the stability of this equilibrium point may change or new equilibrium points may emerge.
You can identify the value of N at which this change occurs by closely analyzing the graph of f(S) and observing the shifts in equilibrium points and population behavior.
(c) If the parameter N slowly and continuously decreases toward the bifurcation value, the population of fox squirrels will exhibit varying behaviors.
Before reaching the bifurcation point, the population may remain relatively stable and fluctuate around an equilibrium point or within a range. However, as N approaches the critical threshold, the population dynamics may become more unpredictable.
Near the bifurcation point, small changes in N can lead to significant changes in the population size and behavior. The population may become more sensitive to disturbances or external factors, resulting in fluctuations or even multiple stable equilibrium points.
Beyond the bifurcation point, the fox squirrel population may exhibit chaotic behavior, where small changes in N or other factors can lead to unpredictable fluctuations and irregular patterns. This behavior is a characteristic of dynamic systems experiencing a bifurcation.
Overall, as N approaches the bifurcation value, the population dynamics become more complex and less predictable, indicating the sensitivity of the system to changes in the carrying capacity.