Math 4791/5791 - Solution Set 5
Fall Semester 1998

  1. (Problem 6.1.3) This system has only one equilibrium point (0,0) and also has the property that it has no linear terms. Therefore linearized stability analysis about the point (0,0) will not work. The only recourse is to plot the vector field. A little analysis shows that

    Notice that these conditions split the phase plane into eight sectors in which the vector field points in different directions (NW, NE, SE or SW). Putting all of this information together gives the first figure at the end of the notes.

  2. (Problem 6.3.1) This system has equilibrium points (2,2) and (-2,-2); the relevant Jacobians are

    displaymath120

    The point (2,2) is an unstable spiral since tex2html_wrap_inline149 and tex2html_wrap_inline151 . The point (-2,-2) is a saddle point since tex2html_wrap_inline149 and tex2html_wrap_inline157 . Further analysis of the saddle reveals that the eigenvalues are tex2html_wrap_inline159 which have eigenvectors tex2html_wrap_inline161 (unstable manifold) and tex2html_wrap_inline163 (stable manifold). Additional analysis yields information about the direction field:

    displaymath121

    With all of this sleuthing done, the phase portrait looks as shown in the second figure at the end of the notes.

  3. (Problem 6.3.4) The equilibrium points are (0,0),(1,0) and (-1,0). Computing the Jacobians, we have

    displaymath122

    Therefore (0,0) is a saddle point with eigenvalues 1 and -1, and eigenvectors tex2html_wrap_inline171 and tex2html_wrap_inline173 respectively. The points tex2html_wrap_inline175 are both stable nodes with eigenvalues -2 and -1, and eigenvectors tex2html_wrap_inline171 and tex2html_wrap_inline179 respectively. The vector field analysis tells us that y'>0 when y<0 and y'<0 when y>0. Along the x-axis (y=0) we also see that x'>0 when x<-1 or 0<x<1, and x'<0 when -1<x<0 or x>1. With all of this information the phase portrait appears as in the last figure at the end of the notes.

  4. (Problem 6.4.2) Since this is a population problem we need to consider only the first quadrant x>0,y>0. There are four equilibrium points: (0,0),(1,1),(0,2),(3/2,0). The Jacobian calculation yields

    displaymath123

    displaymath124

    The point (0,0) is an unstable node with eigenvectors tex2html_wrap_inline171 and tex2html_wrap_inline213 . The point (1,1) is a stable node with eigenvalues tex2html_wrap_inline217 , and eigenvectors tex2html_wrap_inline219 respectively. The other two equilibrium points are saddles: the eigenvalues at (0,2) are 1 and -2 with eigenvectors tex2html_wrap_inline223 and tex2html_wrap_inline213 respectively. The eigenvalues at (3/2,0) are -3 and 1/2 with eigenvectors tex2html_wrap_inline171 and tex2html_wrap_inline231 respectively. The nullclines where x'=0 are x=0 and the line y=2x-3; and the nullclines where y'=0 are y=0 and the line y=2-x. Notice that nullclines intersect at equilibrium points. The nullclines partition the quadrant into four regions in which the direction field has different directions (NW, NE, SE, SW). The figure below (right) shows some direction arrows and some sample trajectories. Notice that as long as the initial conditions are non-zero for both species, the system will approach the stable equilibrium in which both species co-exist. Any initial condition in the basin x>0,y>0 will converge to this equilibrium.

    SEE HARD COPY FOR FIGURE

  5. (Problem 6.4.6) The natural choices for scaling tex2html_wrap_inline247 and tex2html_wrap_inline249 are tex2html_wrap_inline251 and tex2html_wrap_inline253 respectively. Time (t) is most easily scaled to either tex2html_wrap_inline257 or tex2html_wrap_inline259 since these are the natural growth rates in the problem. Choosing the non-dimensional variables tex2html_wrap_inline261 , tex2html_wrap_inline263 and tex2html_wrap_inline265 reduces the original set of ODEs to a system with only three non-dimensional parameters:

    displaymath125

    where

    displaymath126

    Other choices for the parameters are possible, but this choice seems to give the cleanest analysis. Each species is governed by a logistic term which describes intra-species interactions and by a competitive inter-species interaction term. The equilibrium points are (0,0), (0,1), (1,0) which exist for all values of the parameters, plus the point

    displaymath127

    which exists only for certain values of the parameters (see below). The Jacobians are given by

    displaymath128

    The point (0,0) is an unstable node for all parameter values. The point (0,1) is a stable node if tex2html_wrap_inline273 and an (unstable) saddle if tex2html_wrap_inline275 . The point (1,0) is a stable node if tex2html_wrap_inline279 and a saddle if tex2html_wrap_inline281 . Evaluating and analyzing the Jacobian at tex2html_wrap_inline283 is a ponderous task; it is easier to look at the direction field which will tell us what happens near that point.

    The line tex2html_wrap_inline285 is the x-nullcline, while the line tex2html_wrap_inline289 is the y-nullcline. When these lines intersect the equilibrium point tex2html_wrap_inline283 is present. The conditions for this intersection to take place are (i) tex2html_wrap_inline295 and tex2html_wrap_inline281 or (ii) tex2html_wrap_inline273 and tex2html_wrap_inline279 (see figure below). Here is what is happening on the nullclines:

    1. x'=0 and y is decreasing on tex2html_wrap_inline307 when tex2html_wrap_inline307 is above tex2html_wrap_inline311 .
    2. x'=0 and y is increasing on tex2html_wrap_inline307 when tex2html_wrap_inline307 is below tex2html_wrap_inline311 .
    3. y'=0 and x is decreasing on tex2html_wrap_inline311 when tex2html_wrap_inline311 is above tex2html_wrap_inline307 .
    4. y'=0 and x is increasing on tex2html_wrap_inline311 when tex2html_wrap_inline311 is below tex2html_wrap_inline307 .
    With all of this information four different pictures emerge.

    SEE HARD COPY FOR FIGURE

    1. Case 1: tex2html_wrap_inline343 . The only stable equilibrium point is (1,0) which says that species tex2html_wrap_inline247 wins and approaches its carrying capacity (x=1 means tex2html_wrap_inline351 ). This follows also from the biological meaning of the parameters: tex2html_wrap_inline295 means tex2html_wrap_inline355 which in turn means that the natural growth rate of tex2html_wrap_inline247 exceeds the maximum depletion rate of tex2html_wrap_inline247 due to tex2html_wrap_inline249 . Similarly, tex2html_wrap_inline279 means tex2html_wrap_inline365 which in turn means that the natural growth rate of tex2html_wrap_inline249 is less than the maximum depletion rate of tex2html_wrap_inline249 due to tex2html_wrap_inline247 .
    2. Case 2: tex2html_wrap_inline373 . The only stable equilibrium point is (0,1) which says that species tex2html_wrap_inline249 wins and approaches its carrying capacity. The biological interpretation of the parameters is analogous to Case 1.
    3. Case 3: tex2html_wrap_inline373 . In this case neither species fares well in the presence of large numbers of the other species ( tex2html_wrap_inline381 and tex2html_wrap_inline365 ). The equilibrium point tex2html_wrap_inline283 is present but is unstable (a saddle), so co-existence does not occur in this case. Either species can dominate depending on the initial conditions. The dominant species tends towards its carrying capacity.
    4. Case 4; tex2html_wrap_inline387 . Both species are strong in the presence of the other, and the only stable equilibrium state is tex2html_wrap_inline283 (a node) in which both species co-exist at a fraction of their carrying capacities.

      Tue Oct 13 20:51:47 MDT 1998