We continue our discussion on modeling the interactions between populations of predator
and populations of prey.
There is a list here of pertinent literature for
this topic and, one of things we will consider now is a slight modification
to the original Lotka-Volterra model.
We will take into consideration mechanisms of self-regulation in the population of prey.
This may be due to resources that one species,
in this case the prey, needs to take into account for the sustainability.
It could be space, it could be food, it could be exposure to sunlight, really a
number of things, and a known model to take into account basic
elements for self-regulation is known as the logistic model, specifically the logistic
growth.
So we now assume that the population of prey is subject to
logistic growth in the absence of predators and therefore the equations look in this way,
you may compare that to the original formulation in which the population the,
the prey, the growth term for the population of prey
was simply linear.
Now that the term takes, well takes the form that is in red
here in which there is still linear term, rx1, but now there is a negative
minus rX1 squared divided by K. Here in this expression K is called the carrying capacity.
You may think of the carrying capacity as some maximum number of
individuals that population may sustain and we will discuss
next is what about the dynamic behavior, qualitative behavior of
this, of this new model?
We have just made a slight modification to incorporate self-regulation in the equations.
Self-regulation of the population of prey.
Now we have a new set of equations.
Turns out that if we want to compute steady-states, all we have
to do is to set equation number one equals zero, equation number two equals zero, and
solve for both X1 and X2.
We will find two solutions.
One of them is called the trivial solution and we can easily verify those by evaluating
the equations at zero zero, and verify that they become zero.
The nontrivial solutions, an expression that takes this form X1 is mu
or C and X2 equals this expression here, OK?
So we're interested in the stability of the nontrivial steady-state.
We compute the Jacobian matrix, a matrix that contains the
partial derivatives of the right-hand, the right hand side of the equations, excuse me,
and this matrix takes this form when evaluated at
the nontrivial equilibrium point.
The first calculation we do is the calculation of the trace of this matrix.
The trace is defined as the sum of the diagonal entries.
In this case we're adding up this term with zero.
Since we assume that all parameters for this model are positive quantities
then we observe the trace is always negative.
The determinant, the determinant is computed by multiplying this term times zero minus
this term here multiplied by this one and in this case the determinant reduces to
this expression.
So, we have to remember that stability is concluded
or is implied by having a negative trace and a positive determinant.
Another thing to remember is the trace of a matrix
always coincides with the sum of the eigenvalues for that matrix and the determinant coincides
with the product of the eigenvalues, but conditions sort of stability will be in
place when we look for conditions to have a negative trace and a positive
determinant.
In this case the trace is negative so easily we can derive a condition for this
determinant to be positive.
Specifically we observe that if K times c is greater than
mu, then the determinant is positive and the trace is negative which gives
a condition of stability for the nontrivial steady-state.
So let's now verify this conditions of stability by selecting or choosing parameter values
that satisfy such condition and computing numerical solutions.
These are solutions computed, well, the curves you see here are solutions computed
with the R package, L soda.
So, in solid blue, we see the numerical solution for X one of t and dash red is the solution
for X2 of t.
They are displayed versus time.
This is what we call time plots.
You see the value for X1 and X2 at t equals zero.
This is with one set of initial conditions and keeping the same parameter
values and using now a different set of initial conditions we'll, we
calculate or obtain another numerical solution of these equations.
So what we'll, what we will do next, and it's important to remember that for every time
value, say for instance at time equals 80, there
is one value of X1 and a value of X2 so if we now plot these on the
X2X1 plane, we will obtain what is called a phase plane plot
and that's what we will do next.
So, in this plot, in solid black, the curve in solid black is the curve that corresponds
to the upper left panel.
Right?
And you can clearly see what are the initial conditions here for X1 and X2 at time equals
zero.
Another feature of this curve is that there are oscillations
but unlike the original version of the Lotka-Volterra model this oscillations
are damp.
They are damp oscillations instead of sustained oscillations.
Eventually this curves settle or converge.
Their conversion to a value of X1 here and for X2 and that
value at which they converge is the value precisely at which the
solution when displayed in the phase plane plot, this solution is spiraling in and converging
into a value on the phase plane.
That value is the stable steady-state.
OK?
And what we're doing here for purposes of illustration is computing another solution
with a different set of initial conditions that is also displayed.
The solution corresponding to the upper right panel is displayed here in dash, as a dash
curve with color golden, I think, right, so there is a different set of initial conditions.
However, it has a similar qualitative behavior with spiraling in that
are still damp oscillations and eventually it converges to that stable equilibrium point.
Now we will point out that there are other modifications of the original Lotka-Volterra
model similar to the modifications we just reviewed.
For example, we could consider this type of expression, here in red, for
the growth rate for the population of prey.
This involves a parameter R.
This involves a parameter K similar to the carrying
capacity.
It also involves a parameter theta that's acting here as exponent; that exponent is
supposed to be between zero and one.
Here is another example.
It's very similar to the previous with logistic growth.
This is the order here of the terms.
You could also take into consideration making changes on the attack
rate or predation rate instead of the growth rate of the prey.
That, here is an example but I have left in 1961 the predation term now takes this form
and there is another example here of substituting the predation term with an expression that
has a, it's basically a fraction with a numerator, denominator.
This concludes our discussion on some modifications of the Lotka-Volterra
equations.