The Ricker Model If you click on !!! (the menu bar at the top) maple will execute all commands. You can also enter each command as you go along reading this document. You may enter your own commands at any time and place in this document. QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnBsb3RzR0YlISIi 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 The projection function for the Ricker model QyQtSShkaXNwbGF5RzYiNiQtSSVwbG90R0YlNiUtLUkncmlja2VyR0YlNiQkIiImISIiIiQrIjYjSSJQR0YlL0YzOyIiISIkKyUvSSJGR0YlO0Y2IiRdIi1GKDYlLS1GLDYkJCIiKUYwRjFGMkY0RjgiIiI= and the population dynamics QyQtSShkaXNwbGF5RzYiNiM3JC1JLGR5bnBsb3RmaWxtR0YlNiUtSSdyaWNrZXJHRiU2JCQiIiYhIiMiJCsiIiIjIiQrIy1GKTYlRisiJEkiRjMiIiI= The population dynamics for a higher value of r QyQtSSxkeW5wbG90ZmlsbUc2IjYlLUkncmlja2VyR0YlNiQkIiNAISIiIiQrIiIiIyIkKyMiIiI= Ricker models with various values of the parameter r QyQtSShkaXNwbGF5RzYiNiQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShkeW5wbG90R0YlNictSSdyaWNrZXJHRiU2JCwmIiIiRjJJInNHRiUkIiImISIjIiQrIiIiI0Y3IiIhIiQrJS9GMztGOSIjXS9JK2luc2VxdWVuY2VHRiVJJXRydWVHRilGMg== There are two equilibria when r<2. When r<2, 0 is an unstable equilibrium, and the carrying capacity K is a stable equilibrium. We can see this by computing QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJnN0YWIxRzYiNiMtSSdyaWNrZXJHRik2JCQiIz0hIiIiJCsiIiIi or by looking at a cobweb 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 QyQtSShkaXNwbGF5RzYiNiQtSSdjb2J3ZWJHRiU2Jy1JJ3JpY2tlckdGJTYkJCIjPSEiIiIkKyIiIiYiIz8iIiEiJF0iLUYoNidGKiIkPyJGMkYzRjQiIiI= where we see that all population orbits converge to the carrying capacity QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShpdGVyYXRlRzYiNiUtSSdyaWNrZXJHRik2JCQiIz0hIiIiJCsiSSNQMEdGKSIkKyMvRjI3JyIiIiIiJiIiJyIkXSIiJF0jRjY= Let us now consider the case where r>2. We look for a stable 2-cycle for r= QyQsJiIiIkYkLUkiKkclKnByb3RlY3RlZEc2JCIjRCQiIiYhIiNGJEYk which is shown in frame 25 above. QyQ+SSJGRzYiLUkncmlja2VyR0YlNiQkIiREIyEiIyIkKyIiIiI= We first try to solve the equation FF(P)=P, QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJnN0YWIyRzYiNiNJIkZHRikiIiI= Well, it seems to be beyond maple's powers to solve the equation FF(P)=P! Maybe we can solve graphically Let us look at some cobweb diagrams for F^2 QyQtSShkaXNwbGF5RzYiNiYtSSdjb2J3ZWJHRiU2J2YqNiNJIlBHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JIkZHRiU2Iy1GMTYjOSRGJUYlRiUiIzUiIz8iIiEiJCsjLUYoNidGKiIjSUY3RjhGOS1GKDYnRioiIyEpRjdGOEY5LUYoNidGKiIkIT1GN0Y4RjkiIiI= It looks as if there are stable 2-cycles near 50 and 150. We approximate this 2-cycle by orbits: QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShpdGVyYXRlRzYiNiVmKjYjSSJQR0YpRik2JEkpb3BlcmF0b3JHRilJJmFycm93R0YpRiktSSJGR0YpNiMtRjI2IzkkRilGKUYpSSNQMEdGKSIkKyQvRjc3JiIjNSIjSSIjISkiJCE9IiIi and see that we really arrived at a 2-cycle QyQtSSZvcmJpdEc2IjYlSSJGR0YlJkkiJUdGJTYjIiIiIiInRis= We check if it is stable QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JIlBHNiIkIisjUTMiKVslISIpLUklZGlmZkdGJTYkLUkiRkdGLDYjLUY0NiNGK0YrIiIi QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JIlBHNiIkIitpIio9XjohIigtSSVkaWZmR0YlNiQtSSJGR0YsNiMtRjQ2I0YrRisiIiI= Yes, the 2-cycle (44.88, 155.11) is stable. QyQ8Iy1JJHNlcUclKnByb3RlY3RlZEc2JC0mSSZldmFsZkdGJjYjIiIkNiMtSShpdGVyYXRlRzYiNiUtSSdyaWNrZXJHRjA2JCQiIzYhIiIiJCsiSSNQMEdGMCIkKyYvRjk3JyIjNSIjPyIjSSIjWCIjXSIiIg== The stable orbits of the Ricker model can be illustrated in the bifurcation diagram QyQtSSpwb2ludHBsb3RHNiI2JTcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkLUYpNiQ3JCwmJCIjPSEiIiIiIkkiaUdGJSQiIiYhIiQmLUkmb3JiaXRHRiU2JS1JJ3JpY2tlckdGJTYkRi8iJCsiIiMhKSIkXSM2I0kiakdGJS9GQzsiJEkjRkEvRjQ7RjMiJFMjL0kmc3R5bGVHRiVJJnBvaW50R0YlL0kmY29sb3JHRiVJJWJsdWVHRiVGMw== JSFH