Problem $1$: A predator$-$prey model

In class we studied the Lotka$-$Volterra predator$-$prey model which displayed oscillatory solu$-$

tions around the nontrivial fixed point. An unrealistic assumption of that model was that in

the absence of predators, the prey population would grow exponentially without bound.

In this problem we will consider a modification of the Lotka$-$Volterra model to include self$-$

limitation of the prey population.

Consider the system of two differential equations

$\frac{dx}{dt}=rx(1-\frac{x}{M})-$axy,$=f(x,y),$

$\frac{dy}{dt}=bxy-ky,=g(x,y)$;

here $x(t)$ and $y(t)$ are the two population sizes $($which must be $0),$ and $r,M,a,b$ and $k$ are

all positive constants.

$($a$)$ Use the signs of the interaction terms in the DEs to explain why $x(t)$ must be the prey

population size, while $y(t)$ represents the predators.

$($b$)$ This system has five parameters $r,M,a,b$ and $k,$ each of which affects the solution

behaviour. Give a brief biological interpretation of each of these five parameters.

$($c$)$ Find the Jacobian matrix $J(x,y)$ for this system of differential equations.

$($d$)$ Find the nullclines of the system.

$($e$)$ Find the three steady states $($equilibria$/$fixed points$)$ of the system.

Indicate which of the steady states is the trivial equilibrium $($all populations are zero$)$;

which one is a "semi$-$trivial" equilibrium $($only one of the populations is nonzero$)$; and

which is a nontrivial equilibrium $(x^{**},y^{**})($both populations are nonzero, i$.$e$.$ both species

are present; this is a coexistence state$).$

$($f$)$ Depending on the parameter values, the nontrivial equilibrium $(x^{**},y^{**})($with $x^{**},y^{**}0)$

may or may not be biologically realistic $($recall that populations cannot be negative...$).$

Show that the condition for $(x^{**},y^{**})$ to be biologically realistic $-$ that is$,$ for coexistence

to be possible $-$ is that $M>\frac{k}{b}.$

In separate clearly$-$labelled figures, sketch the nullclines in the two cases $M>\frac{k}{b}$ and

$x(x^{**},y^{**})M>\frac{k}{b}r=2M=6,a=1,b=\frac{1}{4}k=1\frac{dx}{dt}=2x(1-\frac{x}{6})-xy,=x(2-\frac{1}{3}x-y),$

$\frac{dy}{dt}=\frac{1}{4}xy-y,=y(\frac{1}{4}x-1).x(0)>0,y(0)>0M$ that the nontrivial steady state $(x^{**},y^{**})is$ not biologically realistic,

what long$-$time behaviour $do$ you expect $in$ this system?

Give a biological interpretation $of$ this situation.

$In$ the remainder $of$ this problem $we$ will explore the case $M>\frac{k}{b}in$ which there $isa$

biologically realistic coexistence equilibrium.

$To$ simplify the calculations, $we$ will choose a particular set $of$ parameter values $r=2,$

$M=6,a=1,b=\frac{1}{4}$ and $k=1$; that $is,in$ the following consider the system

$\frac{dx}{dt}=2x(1-\frac{x}{6})-xy,=x(2-\frac{1}{3}x-y),$

$\frac{dy}{dt}=\frac{1}{4}xy-y,=y(\frac{1}{4}x-1).$

$(h)$ Write down the steady states and nullclines for these particular parameter values, and

determine the direction field $on$ the nullclines.

$(i)$ Use the Jacobian matrix $to$ classify the type and stability $of$ each $of$ the three equilibria.

$(j)$ Draw the phase plane, indicating the steady states, the nullclines, and the direction field

$on$ the nullclines; clearly label all components $of$ your diagram.

Sketch a typical solution trajectory for some nonzero initial condition $(choose$ some small

$x(0)>0,y(0)>0).$

$(k)$ For your solution, describe what happens $to$ the prey and predator populations $in$ the

short and the long term.$M,$ and indicate the locations $of$ all equilibria. Observe how the relative locations

$of$ the nullclines determine whether $or$ not there $is$ a coexistence equilibrium.

$[Hint$: pay careful attention $to$ the intercepts $of$ the $x-$nullcline...$]$

$(g)In$ the case $M$ that the nontrivial steady state $(x^{**},y^{**})is$ not biologically realistic,

what long$-$time behaviour $do$ you expect $in$ this system?

Give a biological interpretation $of$ this situation.

$In$ the remainder $of$ this problem $we$ will explore the case $M>\frac{k}{b}in$ which there $isa$

biologically realistic coexistence equilibrium.

$To$ simplify the calculations, $we$ will choose a particular set $of$ parameter values $r=2,$

$M=6,a=1,b=\frac{1}{4}$ and $k=1$; that $is,in$ the following consider the system

$\frac{dx}{dt}=2x(1-\frac{x}{6})-xy,=x(2-\frac{1}{3}x-y),$

$\frac{dy}{dt}=\frac{1}{4}xy-y,=y(\frac{1}{4}x-1).$

$(h)$ Write down the steady states and nullclines for these particular parameter values, and

determine the direction field $on$ the nullclines.

$(i)$ Use the Jacobian matrix $to$ classify the type and stability $of$ each $of$ the three equilibria.

$(j)$ Draw the phase plane, indicating the steady states, the nullclines, and the direction field

$on$ the nullclines; clearly label all components $of$ your diagram.

Sketch a typical solution trajectory for some nonzero initial condition $(choose$ some small

$x(0)>0,y(0)>0).$

$(k)$ For your solution, describe what happens $to$ the prey and predator populations $in$ the

short and the long term.