2.3 HW 3
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2.3.1 Problem 1
A modification of the predator-prey system is given by
Where is a parameter.
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Find all equilibrium points, in the following two cases: and . (You may select specific
values of , if you wish.)
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Classify the equilibrium points in each case.
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Sketch the nullclines and the phase portraits for different values of .
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What is special about the parameter value ? (It is called a bifurcation value, why?)
solution
2.3.1.1 Part 1
case
Equilibrium points are found by solving
EQ (2) gives or . When then EQ(1) becomes which has solutions . Hence the critical points
found so far are .
When then EQ(1) becomes
Hence or or . Therefore or . Since we are in the case then is positive. Let . Hence . Therefore
the critical points found for this case are .
Hence all the critical points for are For say , these critical points become
case
Equilibrium points are found by solving
EQ (2) gives or . When then EQ(1) becomes which has solutions . Hence the critical points
found so far are .
When then EQ(1) reduces to (as was done above) Hence or . Since we are in the case then is
negative. which means is complex. We are assuming real domain, then these solutions are
rejected. This leaves the critical points for as only the following
2.3.1.2 Part 2
The Jacobian matrix for the system
Is given by the following, where the rule of derivative is used
case
The critical points for this case from part (1) are At point the linearized system matrix is the
Jacobian above evaluated at this point, which gives
Hence becomes
Since the eigenvalues are positive, this is unstable critical point.
At point the linearized system matrix is the Jacobian above evaluated at this point, which gives
Hence becomes
This means this critical points is a saddle point (unstable) since one eigenvalue is negative and
one is negative.
At point the linearized system matrix is the Jacobian above evaluated at this point, which gives
Hence becomes
Therefore
Since then which means the term under the root will remain positive for all values between
and . This means this critical points is a saddle point (unstable) since one eigenvalue will be
negative and one is positive.
At point the linearized system matrix is the Jacobian above evaluated at this point, which gives
Hence becomes
To simplify this, let us pick . Hence the point becomes . The above becomes
Hence and . So one eigenvalue is negative and also the second is negative. This means this is
stable point (positive attraction). Even though we used specific value here, this result is value for
all .
At point the linearized system matrix is the Jacobian above evaluated at this point, which gives
Hence becomes
To simplify this, let us pick . Hence the point becomes . The above becomes
Hence and . So one eigenvalue is negative and also the second is negative. This means this is
stable point (positive attraction). Even though we used specific value here, this result is value for
all .
The following table is a summary of the above results, all for . To obtain numerical values below
for eigenvalues, was used as an example.
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critical point |
eigenvalues |
type of equilibrium |
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negative attraction, unstable |
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saddle point, unstable. |
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| | saddle point, unstable. |
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| | positive attraction, stable |
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positive attraction, stable |
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For specific value the above table becomes
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critical point |
eigenvalues |
type of equilibrium |
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negative attraction, unstable |
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saddle point, unstable. |
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| | saddle point, unstable. |
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| | positive attraction, stable |
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positive attraction, stable |
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case
The critical points for this case from part(1) are At point the linearized system matrix is the
Jacobian above evaluated at this point, which gives
This is the same as with part 1. Which gives double root. Since the eigenvalues are positive,
this is unstable critical point.
At point the linearized system matrix is the Jacobian above evaluated at this point, which gives
This is the same as with part 1. Which gives double root. Since the eigenvalues are positive,
this is unstable critical point.
At point the linearized system matrix is the Jacobian above evaluated at this point, which gives
This is the same as , since is not involved and cancels out. Hence . This means this
critical points is a saddle point (unstable) since one eigenvalue is negative and one is
negative.
At point the linearized system matrix is the Jacobian above evaluated at this point, which gives
From case we found Let us pick . Hence
Therefore this is stable. This is different from case where this point was unstable.
The following table is a summary of the above results, all for . To obtain numerical values below
for eigenvalues, was used as an example.
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critical point |
eigenvalues |
type of equilibrium |
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(same as ) |
negative attraction, unstable |
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| (same as ) | saddle point, unstable. |
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or |
positive attraction, stable |
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From the above, we notice that when changes from to then one critical point switches from
being unstable to stable. This implies solution encountered bifurcation value.
2.3.1.3 Part 3
The nullclines are the solution of and the nullclines are solutions of . Therefore nullclines
are (which is the axis) and . These are both straight lines. To find the nullclines
Hence (which is the axis) and are the nullclines.
The following is a plot of the nullclines for the case of , using
Figure 2.28:nullclines for case
Figure 2.29:code used for the above plot
The following is a plot of the nullclines for the case of , using
Figure 2.30:nullclines for case
We notice that the case of is the following
Figure 2.31:nullclines for case
The following is a phase plot for the case of , using with the critical points highlighted. Red dot
indicates unstable point and green dot color indicates stable point.
Figure 2.32:Phase plot using
Figure 2.33:Code for the above plot
The following is a phase plot for the case of , using
Figure 2.34:Phase plot using
2.3.1.4 Part 4
When the solution of the system changes abruptly. To see this, the following is the two phase
plots about side by side, one for and one for .
Figure 2.35:Phase plots changes as cross over
We notice the following. As changes from to , the equilibrium point changes from being
unstable to stable (this is in addition to now having 3 equilibrium points instead of 5). This is
exactly what bifurcation is. So is a bifurcation value. It is a parameter in the system, which
cause sudden change in the solution trajectories when its value crosses over some specific value,
which is in this problem.
2.3.2 Problem 2
The spread of infectious diseases such as measles, malaria or corona virus may be modeled as
nonlinear system of differential equations, the SIR model. In the simplest form of the model,
we postulate three disjoint groups: , the population of susceptible individuals, , the
infected population, and the recovered population. We assume for simplicity, that
the total population is constant: The SIR model, in its simplest form, is stated as
Where and are parameters.
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Show that the line is an equilibrium line.
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Find the matrix that results from linearizing the system about .
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Calculate the eigenvalues of the resulting matrix.
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Find the nullclines of the system.
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What can we infer about the prospects of full recovery of the population?
solution
This is a diagram of the SIR model
Figure 2.36:SIR model of infection
Hence total population is Which is a constant (This assumes no death occurs, only infection and
recovery). In this model, it is assumed that recovered population can not become infected
again.
2.3.2.1 Part 1
Critical points are solutions to
Eq. (3) shows that, since can not be zero, then is equilibrium. This says that if there are no
infected individuals, then the population do not change which is to be expected since and hence
if then this implies also that and hence remains constant.
2.3.2.2 Part 2
The Jacobian Matrix is Evaluating the above at gives
2.3.2.3 Part 3
To find the eigenvalues
Hence (double root) and are the eigenvalues.
2.3.2.4 Part 4
The nullclines are the solution of and the nullclines are solutions of and nullclines are
solutions of
nullclines are therefore and .
nullclines are therefore and .
nullclines are .
2.3.2.5 Part 5
To answer this, we need to assume that is not zero. This means initial condition such that some
infection exist, otherwise will not change.
Since then as increases (infected population increases) and because then the term
becomes more negative. Since also at the same time becomes smaller during this
process (because more people are infected), then we see that will eventually starts to
decrease as increases and this happens when becomes larger than . (This is the peak
infection).
This means infected population size eventually decreases as passes some peak value. Since
population is assumed constant, this implies the recovered population size will also increase and
eventually all susceptible population that became infected will recover and infected population
will go to zero with time.
2.3.3 Problem 3 (exercise 4.1, page 57)
In exercise 2.3 of chapter 2 we analyzed the existence of periodic solutions in an invariant set of a
three-dimensional system. Obtain this result in a more straightforward manner. The following is
2.3 of chapter 2
We are studying the three-dimensional system
Consider the invariant set . Does this set contain periodic solutions?
solution
Writing the above as
We now need to set before taking the divergence. The above simplifies to Then using
Bendixson’s criterion, periodic solution exist only if divergence of changes sign in the domain or
if the divergence is identically zero. is taken as the whole of .
But
Hence (1) now becomes
Therefore does not change sign. Therefore no periodic solution exist.
2.3.4 key solution for HW 3
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