For the phase portrait shown below find an equation that is

For the phase portrait shown below, find an equation that is consistent with it. here are an infinite number of correct answers-and incorrect ones as well. For each of the following, find an equation x = f(x) with the stated properties, or if there are no examples, explain why not. (In all cases, assume that f(x) is a smooth function Every real number is a fixed point. Every integer is a fixed point, and there are no others. There are precisely three fixed points, and all of them are stable. There are no fixed points. There are precisely 100 fixed points. Recall the Allee effect where, for certain species of organisms the effective growth rate g(N) is highest at an intermediate N and consider the system N = N(r - a, b, r > 0 Find the conditions on a, b, r > 0 where this system provides an example of the Allee effect. Find all the fixed points of the system and classify their stability. Discuss the qualitative difference between this system\'s solutions and those found for the logistic equations (if any).

Solution

Solution 5 : We have given four equilibrium point i.e., -2, 0, 1, 3

So f(x) can be expressed as f(x) = x(x+2)(x-1)(x-3)

Now x\' = f(x) = x(x+2)(x-1)(x-3) is the required equation.