Exponential and logistic growth 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. QyRJKHJlc3RhcnRHJSpwcm90ZWN0ZWRHISIi QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Exponential growth The projection function for logistic growth is QyQ+SSJGRzYiZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JCIiIiomSSJyR0YlRi5GLUYuRi5GJUYlRiVGLg== where r>-1 is a constant. When r>0, F(P)>P and the population is increasing; when -1<r<0, F(P)<P and the populationen is decreasing. Here are the graphs of the projection functions F(P)=P+rP for r=0.05 (increasing population) and -0.05 (decreasing population) together with the diagonal F(P)=P (constant population) QyQtSSVwbG90RzYiNiY3JSomLUkiK0clKnByb3RlY3RlZEc2JCIiIiQiIiYhIiNGLUkiUEdGJUYtKiYtRio2JEYtJCEiJkYwRi1GMUYtRjEvRjE7IiIhIiQrIi9JJ2xhYmVsc0dGJTckSSJGR0YlLUY+NiNGMS9JJmNvbG9yR0YlNyVJJmdyZWVuR0YlRkRJJHJlZEdGJUYt The projection function for the increasing population lies above the projection function for the constant population, and the projectione function for the decreasing population lies below it. We may illustrate the population dymanics by cobwebs QyQtSShkaXNwbGF5RzYiNiQ3JC1JJ2NvYndlYkdGJTYnZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsJiIiIkYzJCIiJiEiI0YzRjM5JEYzRiVGJUYlIiM/IiNLIiIhIiQrIi1GKTYnZipGLEYlRi5GJSomJCIjJipGNkYzRjdGM0YlRiVGJSIjISpGOUY6RjsvSSdsYWJlbHNHRiU3JEYtLUkiRkdGJUYsRjM= The growth function \316\224F(P)=F(P)-P=rP is increasing/decreasing with increasing population density QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiU3JCwkSSJQR0YoJCEiJiEiIywkRiwkIiImRi8vRiw7IiIhIiQrIi9JJ2xhYmVsc0dGKDckRiwtSSkmRGVsdGE7RkdGKDYjRiwiIiI= The population orbit with P0=2 when r=0.05 QyQtSSxkeW5wbG90ZmlsbUc2IjYkZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsJiIiIkYvJCIiJiEiI0YvRi85JEYvRiVGJUYlIiIjRi8= Population orbits with different initial population sizes P0 when r=0.05 QyQtSShkaXNwbGF5RzYiNiUtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShkeW5wbG90R0YlNiVmKjYjSSJQR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUqJiwmIiIiRjYkIiImISIjRjZGNjkkRjZGJUYlRiUsJiIiI0Y2SSJ1R0YlIiUrNSIjXS9GPTsiIiEiIzUvSSdsYWJlbHNHRiU3JEkidEdGJSZGMDYjRkcvSStpbnNlcXVlbmNlR0YlSSZmYWxzZUdGKUY2 The population orbit with P0=2 when r=-0.05 QyQtSSxkeW5wbG90ZmlsbUc2IjYkZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYkIiMmKiEiIyIiIjkkRjFGJUYlRiUiJCsiRjE= Population orbits with different initial population sizes P0 when r=-0.05 QyQtSShkaXNwbGF5RzYiNiUtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShkeW5wbG90R0YlNiVmKjYjSSJQR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUqJiQiIyYqISIjIiIiOSRGOEYlRiVGJSwmIiQrIkY4SSJ1R0YlRjsiI10vRjw7IiIhIiM1L0knbGFiZWxzR0YlNyRJInRHRiUmRjA2I0ZFL0kraW5zZXF1ZW5jZUdGJUkmZmFsc2VHRilGOA== Logistic growth The projection function is QyQ+SSJGRzYiZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JCIiIiooSSJyR0YlRi4sJkYuRi4qJkYtRi5JIktHRiUhIiJGNEYuRi1GLkYuRiVGJUYlRi4= Here is the gaph of the projection function QyQtLUklbG9naUc2IjYkJCIiJiEiIyIkKyI2I0kiUEdGJiIiIg== with r=0.05 and carrying capacity K=100 QyQtSShkaXNwbGF5RzYiNiQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiY3JC0tSSVsb2dpR0YlNiQkIiImISIjIiQrIjYjSSJQR0YlRjcvRjc7IiIhIiUrQS9JInlHRiU7RjoiJCsnL0knbGFiZWxzR0YlNyRGN1ElRihQKUYlLUktaW1wbGljaXRwbG90R0YlNiYvRjdGNS9GNztGOiIlP0EvRj07RjoiJCskL0kmY29sb3JHRiVJJWJsdWVHRiUiIiI= The population is increasing, F(P)>P, when 0<P<K, and decreasing, F(P)<P, when P>K. This may be a little difficult to see on the above graph. It becomes a little more clear if we zoom in on the left part of the graph QyQtSShkaXNwbGF5RzYiNiQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiY3JC0tSSVsb2dpR0YlNiQkIiImISIjIiQrIjYjSSJQR0YlRjcvRjc7IiIhIiQ1Ii9JInlHRiVGOS9JJ2xhYmVsc0dGJTckRjdRJUYoUClGJS1JLWltcGxpY2l0cGxvdEdGJTYmL0Y3RjVGOEY8L0kmY29sb3JHRiVJJWJsdWVHRiUiIiI= Another way of saying this is that the growth function \316\224F(P)=F(P)-P is positive for 0<P<K and negative for P>K. Here is the graph of the growth function The growth function \316\224F(P)=F(P)-P is maximal when the population densiyt P=K/2 is half of the carrying capacity and negative beyond the carrying capacity K=100. QyQtSShkaXNwbGF5RzYiNiM3JC1JJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JSwmLS1JJWxvZ2lHRiU2JCQiIiYhIiMiJCsiNiNJIlBHRiUiIiJGOCEiIi9GODsiIiEiJDUiL0knbGFiZWxzR0YlNyRGOC1JKSZEZWx0YTtGR0YlRjctSS1pbXBsaWNpdHBsb3RHRiU2JjckL0Y4IiNdL0kieUdGJSQiJEQiRjVGOy9GSztGPSQiIzhGOi9JJmNvbG9yR0YlNyRJJWJsdWVHRiVJJmdyZWVuR0YlRjk= The growth function \316\224F(P) is maximal when the population density P=K/2 is half of the carrying capacity. The maximal value of the growth function is called Maximal Sustainable Yield or MSY. The maximal value of the growth function \316\224F(P) is achieved when P=K/2 and it is MSY = max \316\224F(P) =1/4*K*r. In this particular case, MSY = 1.25. JSFH Here are the population orbits at two initial population sizes P0=2 and P0=130 QyQtSShkaXNwbGF5RzYiNiM3JC1JLGR5bnBsb3RmaWxtR0YlNiQtSSVsb2dpR0YlNiQkIiImISIjIiQrIiIiIy1GKTYkRisiJEkiIiIi Population orbits with several different initial population sizes P0 QyQtSShkaXNwbGF5RzYiNiYtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShkeW5wbG90R0YlNiUtSSVsb2dpR0YlNiQkIiImISIjIiQrIiwmIiIjIiIiSSJ1R0YlRjIiJD8iL0Y4OyIiISIjNS9JJ2xhYmVsc0dGJTckSSJQR0YlUSVGKFApRiUvSStpbnNlcXVlbmNlR0YlSSZmYWxzZUdGKS1JJXBsb3RHNiRGKUkoX3N5c2xpYkdGJTYmRjQvSSJ4R0YlO0Y8RjkvSSJ5R0YlO0Y8IiQ1Ii9JJmNvbG9yR0YlSSVibHVlR0YlRjc= When r<2 there are two equilibria. The equilibrium 0 is unstable and the equilibrium K is stable. For instance when r=1.8 we can see this by solving the equation F(P)=P for P and computing the derivative F'(P) at the two solutions QyQtSSZzdGFiMUc2IjYjLUklbG9naUdGJTYkJCIjPSEiIiIkKyIiIiI= or by drawing cobweb diagrams QyQtSShkaXNwbGF5RzYiNiQtSSdjb2J3ZWJHRiU2Jy1JJWxvZ2lHRiU2JCQiIz0hIiIiJCsiIiIjIiM6IiIhIiQ/Ii1GKDYnRipGNCIjNUYzRjQiIiI= The stable equilibrium determines the long term behaviou becasuse all population orbits (from not too big initial populations) converge to the carrying capacity QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSZmbG9vckc2JEYlSShfc3lzbGliRzYiNiMtSShpdGVyYXRlR0YrNiUtSSVsb2dpR0YrNiQkIiM9ISIiIiQrIkkjUDBHRisiJCsjL0Y3NyciIiIiIiYiIiciJF0iIiRdI0Y7 Let us now consider what happens when r>2. Here are four examples with increasing parameter r. QyQtSSxkeW5wbG90ZmlsbUc2IjYkLUklbG9naUdGJTYkJCIjQSEiIiIkKyIiIiMiIiI= QyQtSSxkeW5wbG90ZmlsbUc2IjYkLUklbG9naUdGJTYkJCIjRCEiIiIkKyIiIiMiIiI= QyQtSSxkeW5wbG90ZmlsbUc2IjYkLUklbG9naUdGJTYkJCIjRSEiIiIkKyIiIiMiIiI= QyQtSSxkeW5wbG90ZmlsbUc2IjYkLUklbG9naUdGJTYkJCIjRiEiIiIkKyIiIiMiIiI= Logistic growth with varying parameter r - stable cycles and chaos emerge QyQtSShkaXNwbGF5RzYiNiQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShkeW5wbG90R0YlNictSSVsb2dpR0YlNiQsJiIiIkYySSJzR0YlJCIiJiEiIyIkKyIiI11GNyIiISIkXSIvRjM7RjkiI1MvSStpbnNlcXVlbmNlR0YlSSV0cnVlR0YpRjI= When r is slightly bigger than 2 stable cycles occur. We look for 2-cycles in logi(2.2,100). QyQ+SSJGRzYiLUklbG9naUdGJTYkJCIjQSEiIiIkKyIiIiI= The population dynamics with P0=5 indicates that there is a 2-cycle QyQtSShkeW5wbG90RzYiNiVJIkZHRiUiIiYiJCsiIiIi We can find the 2-cycle by solving the equation F(F(P))=P for P. We also compute the derivative (FF)'(P) at each solution to determine stability. QyQtSSZzdGFiMkc2IjYjSSJGR0YlIiIi Alternatively, we can use the cobweb diagram for FF QyQtSShkaXNwbGF5RzYiNiQtSSdjb2J3ZWJHRiU2J2YqNiNJIlBHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JIkZHRiU2Iy1GMTYjOSRGJUYlRiUiI1MiIz8iIiEiJEkiLUYoNidGKiIjXUY3RjhGOSIiIg== It looks as if there are stable 2-cycles through 70 and 115 and unstable 2-cycles at 0 and 100. The stable 2-cycles determine the long term behavior because population orbits will converge to them. We compute the 100th generation F^100(P0) with four different initial populations QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShpdGVyYXRlRzYiNiVmKjYjSSJQR0YpRik2JEkpb3BlcmF0b3JHRilJJmFycm93R0YpRiktSSJGR0YpNiMtRjI2IzkkRilGKUYpSSNQMEdGKSIkKyIvRjc3JiIjUyIjXSIjISkiJD8iIiIi and check that we indeed have a 2-cycle QyQtSSZvcmJpdEc2IjYlSSJGR0YlJkkiJUdGJTYjIiIiIiInRis= and that it is stable QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JIlBHNiIkIiszY1lpdSEiKS1JJWRpZmZHRiU2JC1JIkZHRiw2Iy1GNDYjRitGKyIiIg== QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JIlBHNiIkIitcViVHOyIhIigtSSVkaWZmR0YlNiQtSSJGR0YsNiMtRjQ2I0YrRisiIiI= We next look for 4-cycles in logi(2.5,100); QyQ+SSJGRzYiLUklbG9naUdGJTYkJCIjRCEiIiIkKyIiIiI= QyQtSShkeW5wbG90RzYiNiVJIkZHRiUiIiMiI10iIiI= There seems to be a stable 4 cycle near 53. Any orbit near 53 will converge to it. QyQtSShpdGVyYXRlRzYiNiVJIkZHRiUiI2IiJCsjIiIi We check that this is indeed a 4-cycle QyQtSSZvcmJpdEc2IjYlSSJGR0YlSSIlR0YlIiIlIiIi and that it is stable QyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kiUEc2IiQiK2piWmZgISIpLUklZGlmZkdGJTYkLUkiRkdGKTYjLUYxNiMtRjE2Iy1GMTYjRihGKCIiIg== We could also try to compute it directly or use cobweb QyQtSSZzdGFiNEc2IjYjLUklbG9naUdGJTYkJCIjRCEiIiIkKyIiIiI= QyQtSShkaXNwbGF5RzYiNiUtSSdjb2J3ZWJHRiU2J2YqNiNJIlBHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JIkZHRiU2Iy1GMTYjLUYxNiMtRjE2IzkkRiVGJUYlIiM/RjoiIiEiJEkiLUYoNidGKiIjWEY6RjtGPC1GKDYnRioiIyMpRjpGO0Y8IiIi We approximate the stable cycles by orbits QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShpdGVyYXRlRzYiNiVmKjYjSSJQR0YpRik2JEkpb3BlcmF0b3JHRilJJmFycm93R0YpRiktSSJGR0YpNiMtRjI2Iy1GMjYjLUYyNiM5JEYpRilGKUkjUDBHRikiJCsiL0Y7NyYiIz8iI1giIyMpIiQ1IiIiIg== QyQtSSZvcmJpdEc2IjYlSSJGR0YlJCIrXSp5QiwoISIpIiIpIiIi This is the 4-cycle we found before. We have now also found also another cycle, which is a 2-cycle QyQtSSZvcmJpdEc2IjYlLUklbG9naUdGJTYkJCIjRCEiIiIkKyIkIisrKysrZyEiKSIiKSIiIg== but it is unstable QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JIlBHNiIiI2ctSSVkaWZmR0YlNiQtSSJGR0YsNiMtRjI2Iy1GMjYjLUYyNiNGK0YrIiIi QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXN1YnNHRiU2JC9JIlBHNiIiI2ctSSVkaWZmR0YlNiQtSSJGR0YsNiMtRjI2I0YrRisiIiI= The unstability can be illustrated by orbits through points close to P0=20 QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSShpdGVyYXRlRzYiNiVJIkZHRilJI1AwR0YpIiQrIi9GLDclJCImJioqPiEiJCIjPyQiJjArI0YyIiIi Here is the cobweb diagram with r=2.5: QyQtSSdjb2J3ZWJHNiI2Jy1JJWxvZ2lHRiU2JCQiI0QhIiIiJCsiIiNdIiUrNSIiISIkSSIiIiI= and here is the cobweb diagram with r=2.6: QyQtSSdjb2J3ZWJHNiI2Jy1JJWxvZ2lHRiU2JCQiI0UhIiIiJCsiIiNdIiUrNSIiISIkSSIiIiI= The stable orbits are indicated in the bifurcation diagram QyQtSSpwb2ludHBsb3RHNiI2JTcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkLUYpNiQ3JCwmJCIjPSEiIiIiIkkiaUdGJSQiIiYhIiQmLUkmb3JiaXRHRiU2JS1JJWxvZ2lHRiU2JEYvIiQrIiIjISkiJF0jNiNJImpHRiUvRkM7IiRJI0ZBL0Y0O0YzIiRTIy9JJnN0eWxlR0YlSSZwb2ludEdGJS9JJmNvbG9yR0YlSSVibHVlR0YlRjM= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Logistic growth with harvest - try to change the harvest QyQtSSxkeW5wbG90ZmlsbUc2IjYkLCYtSSVsb2dpR0YlNiQkIiImISIiIiQrIiIiIiQhJDAiRi1GLyIjSUYv QyQtSShkeW5wbG90RzYiNiUsJi1JJWxvZ2lHRiU2JCQiIiYhIiIiJCsiIiIiISIlRi8iIz9GLkYv Maximal Sustainable Yield MSY QyQ+SSJGRzYiLUklbG9naUdGJTYkJCIiKSEiIiIkKyMiIiI= QyQ+SSRNU1lHNiIsJC1JIipHJSpwcm90ZWN0ZWRHNiQjIiIiIiIlIiQrIyQiIikhIiJGLA== An equilibrium for logistic growth with harvest H is a population size P so that growth \316\224F(P) equals harvest, H. The equilibria can be determined from the intersection points between the graph of the growth function and the horizontal line at level H QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQ3JywmLUkiRkdGKDYjSSJQR0YoIiIiRi8hIiJJJE1TWUdGKCwkRjIkRjBGMSwkRjIkIiImRjEsJEYyJCIiKUYxL0YvOyIiISIkKyNGMA== Let us consider the case where the harvest is QyQ+SSJIRzYiLCRJJE1TWUdGJSQiIiYhIiIiIiI= The equilibria are QyQtSSZzdGFiMUc2IjYjZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYtSSJGR0YlNiM5JCIiIkkiSEdGJSEiIkYlRiVGJUYy The smallest equilibrium P1=29 is unstable, the biggest P2=170 is stable. This means that if the population size is less than 29, the population will go extinct; if the population is more than 29, it will in the long term stabilize at population size 170. Here are the population orbits from P0=28, P0=30 and P0=50 QyQtSShkaXNwbGF5RzYiNiM3Ji1JKGR5bnBsb3RHRiU2JWYqNiNJIlBHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmLUkiRkdGJTYjOSQiIiJJIkhHRiUhIiJGJUYlRiUiI0kiIz8tRik2JUYrIiNdRjotRik2JUYrIiNHIiImLUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlJCIlR0ghIiMvRi07IiIhRjovSSZjb2xvckdGJUklYmx1ZUdGJUY2 We can also see this from the cobweb diagrams from P0=28 and P0=30 QyQtSShkaXNwbGF5RzYiNiM3JC1JJ2NvYndlYkdGJTYnZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYtSSJGR0YlNiM5JCIiIkkiSEdGJSEiIkYlRiVGJSIjRyIiKCIiISIkIT0tRik2J0YrIiNJIiM/RjtGPEY2 JSFH