2.3 HW 3

  2.3.1 Problem 1
  2.3.2 Problem 2
  2.3.3 Problem 3 (exercise 4.1, page 57)
  2.3.4 key solution for HW 3
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2.3.1 Problem 1

   2.3.1.1 Part 1
   2.3.1.2 Part 2
   2.3.1.3 Part 3
   2.3.1.4 Part 4

A modification of the predator-prey system is given byx˙=x(1x)axyx+1y˙=y(1y)

Where a>0 is a parameter.

  1. Find all equilibrium points, in the following two cases: 0<a<1 and a>1. (You may select specific values of a, if you wish.)
  2. Classify the equilibrium points in each case.
  3. Sketch the nullclines and the phase portraits for different values of a.
  4. What is special about the parameter value a=1? (It is called a bifurcation value, why?)

solution

2.3.1.1 Part 1

case 0<a<1

Equilibrium points are found by solving (1)x(1x)axyx+1=0(2)y(1y)=0

EQ (2) gives y=0 or y=1. When y=0 then EQ(1) becomes x(1x)=0 which has solutions x=0,x=1. Hence the critical points found so far are (0,0),(1,0).

When y=1 then EQ(1) becomes x(1x)axx+1=0x(1x)(1+x)ax=0xx3ax=0x(1x2a)=0

Hence x=0 or 1x2a=0 or x21+a=0. Therefore x2=1a or x=±1a. Since we are in the case 0<a<1 then 1a is positive. Let 1a=n2. Hence x=±n. Therefore the critical points found for this case are (0,1),(1a,1),(1a,1).

Hence all the critical points for 0<a<1 are(xi,yi)={(0,0),(1,0),(0,1),(1a,1),(1a,1)} For say a=12, these critical points become(xi,yi)={(0,0),(1,0),(0,1),(12,1),(12,1)}={(0,0),(1,0),(0,1),(0.707107,1),(0.707107,1)}

case a>1

Equilibrium points are found by solving (1)x(1x)axyx+1=0(2)y(1y)=0

EQ (2) gives y=0 or y=1. When y=0 then EQ(1) becomes x(1x)=0 which has solutions x=0,x=1. Hence the critical points found so far are (0,0),(1,0).

When y=1 then EQ(1) reduces to (as was done above) x(1x2a)=0 Hence x=0 or x=±1a. Since we are in the case a>1 then 1a is negative. which means 1a is complex. We are assuming real domain, then these solutions are rejected. This leaves the critical points for a>1 as only the following(xi,yi)={(0,0),(1,0),(0,1)}

2.3.1.2 Part 2

The Jacobian matrix for the system x˙=x(1x)axyx+1y˙=y(1y)

Is given by the following, where the rule of derivative ddxf(x)g(x)=gffgg2 is usedJ=(x˙xx˙yy˙xy˙y)=((1x)x(x+1)ay(axy)(1+x)2axx+10(1y)y)=(12xaxy+ayaxy(1+x)2axx+1012y)=(12xay(1+x)2axx+1012y)

case 0<a<1

The critical points for this case from part (1) are(xi,yi)={(0,0),(1,0),(0,1),(1a,1),(1a,1)} At point (0,0) the linearized system A matrix is the Jacobian above evaluated at this point, which givesA=(12xay(1+x)2axx+1012y)=(1001)

Hence |AλI|=0 becomes|1λ001λ|=0(1λ)2=0λ=1 double root

Since the eigenvalues are positive, this is unstable critical point.

At point (1,0) the linearized system A matrix is the Jacobian above evaluated at this point, which givesA=(12xay(1+x)2axx+1012y)=(12a201)=(1a201)

Hence |AλI|=0 becomes|1λa201λ|=0(1λ)(1λ)=0λ21=0λ2=1λ=±1

This means this critical points is a saddle point (unstable) since one eigenvalue is negative and one is negative.

At point (0,1) the linearized system A matrix is the Jacobian above evaluated at this point, which givesA=(12xay(1+x)2axx+1012y)=(1a001)

Hence |AλI|=0 becomes|(1a)λ001λ|=0((1a)λ)(1λ)=0λ2+aλ+a1=0

Therefore λ=b2a±12ab24ac=a2±12a24(a1)=a2±12a24a+4

Since 0<a<1 then 3<a24a<0 which means the term under the root will remain positive for all a values between 0 and 1. This means this critical points is a saddle point (unstable) since one eigenvalue will be negative and one is positive.

At point (1a,1) the linearized system A matrix is the Jacobian above evaluated at this point, which givesA=(12xay(1+x)2axx+1012y)=(121aa(1+1a)2a1a1a+101)

Hence |AλI|=0 becomes|(121aa(1+1a)2)λa1a1a+101λ|=0((121aa(1+1a)2)λ)(1λ)=0

To simplify this, let us pick a=12. Hence the point (1a,1) becomes (0.707,1). The above becomes((121212(1+12)2)λ)(1λ)=0(λ+1)(λ2+2)=0

Hence λ=1 and λ=22. So one eigenvalue is negative and also the second is negative. This means this is stable point (positive attraction). Even though we used specific a value here, this result is value for all 0<a<1.

At point (1a,1) the linearized system A matrix is the Jacobian above evaluated at this point, which givesA=(12xay(1+x)2axx+1012y)=(1+21aa(11a)2a1a1a+101)

Hence |AλI|=0 becomes|(1+21aa(11a)2)λa1a1a+101λ|=0((1+21aa(11a)2)λ)(1λ)=0

To simplify this, let us pick a=12. Hence the point (1a,1) becomes (0.707,1). The above becomes((1+21212(112)2)λ)(1λ)=0(λ+1)(λ+2+2)=0

Hence λ=1 and λ=22. So one eigenvalue is negative and also the second is negative. This means this is stable point (positive attraction). Even though we used specific a value here, this result is value for all 0<a<1.

The following table is a summary of the above results, all for 0<a<1. To obtain numerical values below for eigenvalues, a=12 was used as an example.




critical point eigenvalues type of equilibrium






(0,0) λ1=1,λ2=1 negative attraction, unstable



(1,0) λ=±1 saddle point, unstable.



(0,1) λ=a2±12a24a+4 saddle point, unstable.



(1a,1) λ1=1,λ2=22 positive attraction, stable



(1a,1) λ1=1,λ2=22 positive attraction, stable



For specific value a=12 the above table becomes




critical point eigenvalues type of equilibrium






(0,0) λ1=1,λ2=1 negative attraction, unstable



(1,0) λ=±1 saddle point, unstable.



(0,1) λ1=1,λ2=12 saddle point, unstable.



(0.707,1) λ1=1,λ2=0.5857 positive attraction, stable



(0.707,1) λ1=1,λ2=3.41421 positive attraction, stable



case a>1

The critical points for this case from part(1) are(xi,yi)={(0,0),(1,0),(0,1)} At point (0,0) the linearized system A matrix is the Jacobian above evaluated at this point, which givesA=(12xay(1+x)2axx+1012y)=(1001)

This is the same as with part 1. Which gives λ=1 double root.  Since the eigenvalues are positive, this is unstable critical point.

At point (1,0) the linearized system A matrix is the Jacobian above evaluated at this point, which givesA=(12xay(1+x)2axx+1012y)=(1001)

This is the same as with part 1. Which gives λ=1 double root.  Since the eigenvalues are positive, this is unstable critical point.

At point (1,0) the linearized system A matrix is the Jacobian above evaluated at this point, which givesA=(12xay(1+x)2axx+1012y)=(1a201)

This is the same as 0<a<1, since a is not involved and cancels out. Hence λ=±1. This means this critical points is a saddle point (unstable) since one eigenvalue is negative and one is negative.

At point (0,1) the linearized system A matrix is the Jacobian above evaluated at this point, which givesA=(12xay(1+x)2axx+1012y)=(1a001)

From case 0<a<1 we foundλ=a2±12a24a+4 Let us pick a=3. Henceλ=32±12912+4=32±12=2,1

Therefore this is stable. This is different from case 0<a<1 where this point was unstable.

The following table is a summary of the above results, all for a>1. To obtain numerical values below for eigenvalues, a=3 was used as an example.




critical point eigenvalues type of equilibrium






(0,0) λ1=1,λ2=1 (same as 0<a<1) negative attraction, unstable



(1,0) λ=±1 (same as 0<a<1) saddle point, unstable.



(0,1) λ=a2±12a24a+4 or λ1=1,λ2=2 positive attraction, stable



From the above, we notice that when a changes from 0<a<1 to a>1 then one critical point (0,1) switches from being unstable to stable. This implies solution encountered bifurcation value.

2.3.1.3 Part 3

x˙=x(1x)axyx+1y˙=y(1y)

The x nullclines are the solution of x(1x)axyx+1=0 and the y nullclines are solutions of y(1y)=0. Therefore y nullclines are y=0 (which is the x axis) and y=1. These are both straight lines. To find the x nullclinesx(1x)axyx+1=0x(1x)(x+1)axy=0xx3axy=0x(1x2ay)=0

Hence x=0 (which is the y axis) and x2=1ay are the x nullclines.



x nullclines y nullclines




x=0 y=0


x2=1ay y=1


The following is a plot of the nullclines for the case of 0<a<1, using a=12

pict
Figure 2.28:nullclines for case 0<a<1

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Figure 2.29:code used for the above plot

The following is a plot of the nullclines for the case of a>1, using a=3

pict
Figure 2.30:nullclines for case a>1

We notice that the case of a=1 is the following

pict
Figure 2.31:nullclines for case a>1

The following is a phase plot for the case of 0<a<1, using a=12 with the critical points highlighted. Red dot indicates unstable point and green dot color indicates stable point.

pict
Figure 2.32:Phase plot 0<a<1 using a=0.5

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Figure 2.33:Code for the above plot

The following is a phase plot for the case of a>1, using a=3.

pict
Figure 2.34:Phase plot a>1 using a=3
2.3.1.4 Part 4

When a=1 the solution of the system changes abruptly. To see this, the following is the two phase plots about side by side, one for a=0.99 and one for a=1.01.

pict
Figure 2.35:Phase plots changes as a cross over 1

We notice the following. As a changes from 0<a<1 to a>1, the equilibrium point (0,1) 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 a=1 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 1 in this problem.

2.3.2 Problem 2

   2.3.2.1 Part 1
   2.3.2.2 Part 2
   2.3.2.3 Part 3
   2.3.2.4 Part 4
   2.3.2.5 Part 5

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: S=S(t), the population of susceptible individuals, I=I(t), the infected population, and R=R(t) the recovered population. We assume for simplicity, that the total population is constant:ddt(S+I+R)=0 The SIR model, in its simplest form, is stated as(1)S˙=βSI(2)I˙=βSIυI(3)R˙=υI

Where υ>0 and β>0 are parameters.

  1. Show that the line I=0 is an equilibrium line.
  2. Find the matrix that results from linearizing the system about I=0.
  3. Calculate the eigenvalues of the resulting matrix.
  4. Find the nullclines of the system.
  5. What can we infer about the prospects of full recovery of the population?

solution

This is a diagram of the SIR model

pict
Figure 2.36:SIR model of infection

Hence total population is N=S+I+R Which is a constant (This assumes no death occurs, only infection and recovery). In this model, it is assumed that recovered population R(t) can not become infected again.

2.3.2.1 Part 1

Critical points are solutions to (1A)0=βSI(2A)0=βSIυI(3A)0=υI

Eq. (3) shows that, since υ can not be zero, then I=0 is equilibrium. This says that if there are no infected individuals, then the population N do not change which is to be expected since N=S+I+R and hence if I=0 then this implies also that R=0 and hence N remains constant.

2.3.2.2 Part 2

The Jacobian Matrix isJ=(S˙SS˙IS˙RI˙SI˙II˙RR˙SR˙IR˙R)=(βIβ0βIβSυ00υ0) Evaluating the above at I=0 givesA=(0β00βSυ00υ0)

2.3.2.3 Part 3

To find the eigenvalues |AλI|=0|λβ00(βSυ)λ00υλ|=0λ|(βSυ)λ0υλ|+β|000λ|+0=0λ2(βSυλ)=0

Hence λ=0 (double root) and λ=βSυ are the eigenvalues.

2.3.2.4 Part 4

The S nullclines are the solution of βSI=0 and the I nullclines are solutions of βSIυI=0 and R nullclines are solutions of υI

S nullclines are therefore I=0 and S=0.

I nullclines are therefore I=0 and S=υβ.

R nullclines are I=0.

2.3.2.5 Part 5

To answer this, we need to assume that I(0) is not zero. This means initial condition such that some infection exist, otherwise S(t) will not change.

Since I˙=βSIυI then as I increases (infected population increases) and because υ>0 then the term υI becomes more negative. Since S(t) also at the same time becomes smaller during this process (because more people are infected), then we see that I˙ will eventually starts to decrease as I increases and this happens when υI becomes larger than βSI. (This is the peak infection).

This means infected population size eventually decreases as I(t) 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 systemx˙1=x1x1x2x23+x3(x12+x221x1+x1x2+x23)x˙2=x1x3(x1x2+2x1x2)x˙3=(x31)(x3+2x3x22+x33)

Consider the invariant set x3=1. Does this set contain periodic solutions?

solution

Writing the above as x˙=F(x1,x2,x2)(x˙1x˙2x˙3)=(f1(x1,x2,x3)f2(x1,x2,x3)f3(x1,x2,x3))=(x1x1x2x23+x3(x12+x221x1+x1x2+x23)x1x3(x1x2+2x1x2)(x31)(x3+2x3x22+x33))

We now need to set x3=1 before taking the divergence. The above simplifies to(x˙1x˙2x˙3)=(f1(x1,x2,x3)f2(x1,x2,x3)f3(x1,x2,x3))=(x1x1x2x23+(x12+x221x1+x1x2+1)x1(x1x2+2x1x2)0) Then using Bendixson’s criterion, periodic solution exist only if divergence of F(x1,x2,x2) changes sign in the domain D or if the divergence is identically zero. D is taken as the whole of R3. (1)F(x1,x2,x2)=f1x1+f2x2+f3x3 But f1x1=1x2+(2x11+x2)=2x1f2x2=(1+2x1)=12x1f3x3=0

Hence (1) now becomes F(x1,x2,x2)=2x1+12x1=1

Therefore F(x1,x2,x2) does not change sign. Therefore no periodic solution exist.

2.3.4 key solution for HW 3

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