Modeling Population Dynamics of Flies, Frogs, and Crocodiles

[Jbreezy]

Homework Statement

Flies, frogs, and crocodiles coexist in an environment. To survive, frogs need to eat flies and crocodiles need to eat frogs. In the absence of frogs, the fly population will grow exponentially and the crocodile population will decay exponentially. In the absence of crocodiles and flies, the grog population will decay exponentially. If P(t) , Q(t), and R(t) represent the populations of these three species at time t, write a system of differential equations as a model for their evolution. If the constants in your equation are all positive, explain why you have used plus or minus signs.

Homework Equations

Not really sure how to do this or where to start at.

The Attempt at a Solution

I said Dp/dt is for my crocs so p is that variable to represent crocidiles.
I said dq/dt is for frogs so q is the variable for frogs and dr/dt is for flies so r is for flies.

I'm having trouble to get going here.

 

[pasmith]

Homework Statement

Flies, frogs, and crocodiles coexist in an environment. To survive, frogs need to eat flies and crocodiles need to eat frogs. In the absence of frogs, the fly population will grow exponentially and the crocodile population will decay exponentially. In the absence of crocodiles and flies, the grog population will decay exponentially. If P(t) , Q(t), and R(t) represent the populations of these three species at time t, write a system of differential equations as a model for their evolution. If the constants in your equation are all positive, explain why you have used plus or minus signs.

Homework Equations

Not really sure how to do this or where to start at.

The Attempt at a Solution

I said Dp/dt is for my crocs so p is that variable to represent crocidiles.
I said dq/dt is for frogs so q is the variable for frogs and dr/dt is for flies so r is for flies.

I'm having trouble to get going here.

You are looking for a linear system with constant coefficients, where
<br /> \dot P = a_1 P + a_2 Q + a_3 R \\<br /> \dot Q = b_1 P + b_2 Q + b_3 R \\<br /> \dot R = c_1 P + c_2 Q + c_3 R <br />
Now use the given assumptions to determine which coefficients are positive, which are negative, and which are zero.

 

[Jbreezy]

You are looking for a linear system with constant coefficients, where
<br /> \dot P = a_1 P + a_2 Q + a_3 R \\<br /> \dot Q = b_1 P + b_2 Q + b_3 R \\<br /> \dot R = c_1 P + c_2 Q + c_3 R <br />
Now use the given assumptions to determine which coefficients are positive, which are negative, and which are zero.

But why would the system be linear when in the directions it says that some of them fall die off exponentially or grow exponentially? Just wondering

 

[Mandelbroth]

But why would the system be linear when in the directions it says that some of them fall die off exponentially or grow exponentially? Just wondering

Suppose we have $f'(x)=a_1f(x)$. What is $f$?

 

[Jbreezy]

So like if you integrated it ? I don't get it. Or just f(x) = f(x)' / a1 ??

 

[Mandelbroth]

So like if you integrated it ? I don't get it. Or just f(x) = f(x)' / a1 ??

Try $a_1=\frac{f'(x)}{f(x)}$. This is a differential equation you should be able to solve.

 

[Jbreezy]

This is what I came up with I doubt it is right.

$\dot P = a_1 P + a_2 Q + a_3 R \\ \dot Q = b_1 P + b_2 Q - b_3 R \\ \dot R = -c_1 P - c_2 Q + c_3 R$
In order of equations;
because crocs decay exponentially if frogs are gone and the flies will grow.
all pos. because if all are working crocs are happy.
Minus b_3 because if there are no crocs or flies the frogs decay
because crocs decay exponentially if frogs are gone and the flies will grow.

 

[haruspex]

This is what I came up with I doubt it is right.

$\dot P = a_1 P + a_2 Q + a_3 R \\ \dot Q = b_1 P + b_2 Q - b_3 R \\ \dot R = -c_1 P - c_2 Q + c_3 R$
In order of equations;
because crocs decay exponentially if frogs are gone and the flies will grow.
all pos. because if all are working crocs are happy.
Minus b_3 because if there are no crocs or flies the frogs decay
because crocs decay exponentially if frogs are gone and the flies will grow.

I'm not sure which label represents which species. The OP order is flies, frogs, crocs, so I'll assume that's P, Q, R respectively.
Consider the case of no frogs, Q(t) = 0 for all t. Will the fate of the crocs then depend in any way on the number of flies (and vice versa)? What does that tell you about the coefficients?

 

[Jbreezy]

$\dot P = a_1 P + a_2 Q + a_3 R \\ \dot Q = b_1 P + b_2 Q - b_3 R \\ \dot R = -c_1 P - c_2 Q + c_3 R$



OK your order is correct Haruspex. I redid my equations. I think that the coefficents in those equation should be 0?

$\dot P = a_1 P - a_2 Q \\ \dot Q = b_1 P - b_2 Q + b_3 R \\ \dot R = c_2 Q - c_3 R$
I was just thinking that the flies and crocs are only related through the frogs.

 

[D H]

Think about how crocs affect the frog population. Your equation for $\dot Q$ has the frog population growing as the croc population grows ever larger. Does that make any sense?

For example, suppose a1=a2=P=0, b2=c3=1, and b3=c2=2. There are no flies, yet the population of frogs and crocs grow exponentially.

 

[Jbreezy]

Think about how crocs affect the frog population. Your equation for $\dot Q$ has the frog population growing as the croc population grows ever larger. Does that make any sense?

For example, suppose a1=a2=P=0, b2=c3=1, and b3=c2=2. There are no flies, yet the population of frogs and crocs grow exponentially.

R is crocs. So in my second equation I have ...

$\dot P = a_1 P - a_2 Q \\ \dot Q = b_1 P - b_2 Q + b_3 R \\ \dot R = c_2 Q - c_3 R$

Frogs are negative in the equation for Q so doesn't that mean they decrease? Oh man I'm confused.

 

[D H]

Why is the relation between increasing frog population ($\dot Q$) and croc population ($b_3 R$) a positive one? Does that make *any* sense? What happens to the poor frogs if there are a lot of crocs around?

 

[haruspex]

You are looking for a linear system with constant coefficients

Of course, a linear system will exhibit some unrealistic behaviour. E.g. if R = 0 at some time but Q > 0 then the crocs will come back from extinction.

 

[pasmith]

Of course, a linear system will exhibit some unrealistic behaviour. E.g. if R = 0 at some time but Q > 0 then the crocs will come back from extinction.

I would prefer a non-linear system (there really ought to be a non-trivial fixed point corresponding to crocodiles being extinct).

Of course, once a population is sufficiently small the assumption that it can be modeled as a real-valued function of time no longer holds.

 

[haruspex]

I would prefer a non-linear system (there really ought to be a non-trivial fixed point corresponding to crocodiles being extinct).

Of course, once a population is sufficiently small the assumption that it can be modeled as a real-valued function of time no longer holds.

Right, but even before that there is the doubtful assumption that the fecundity is directly proportional to the prey/predator ratio and with no time lag.

 

[mfb]

Right, but even before that there is the doubtful assumption that the fecundity is directly proportional to the prey/predator ratio and with no time lag.

Even without taking time lag into account, the frogs don't produce crocs. Crocs have to multiply on their own, and this will need crocs.

Why are we looking for a linear system?

 

[haruspex]

Even without taking time lag into account, the frogs don't produce crocs. Crocs have to multiply on their own, and this will need crocs.

Yes, that was the point previously discussed. I was noting that even when there are lots of crocs, if there are humungous numbers of frogs then the simple linear relationship will break down.
I agree it is not given that a linear relationship should be assumed, but it does fit with information given about exponential growth and decay.

 

[mfb]

The given exponential parts are fine (not completely realistic, but good enough), but that does not say anything about the interaction terms.