Hello! I've gotten stuck on an assignment which i need to turn in as soon as possible, so any help is very appreciated.
The assignment is simply to solve a problem concerning the Lotka-Volterra prey predator model, but - to no ones surprise - there is a twist: it includes intraspecies conflict, meaning that there's a couple more coefficients than I've commonly found on webbsites, our textbook or youtube (but I might have missed something obvious). Also, one of the coefficients (c) has a negative sign that gives the nullcline for y a positive incline... The problem is as follows (alternatively, see the attached image):
---------------------------------------------------
Let \(\displaystyle x(t)\) and \(\displaystyle y(t)\) be the populations of prey and predator at time \(\displaystyle t\, \geq\, 0,\) respectively.
Consider the following model:
. . . . .\(\displaystyle \dfrac{d}{dt}\, x(t)\, =\, x(t)\left(a\, -\, ex(t)\, -\, by(t)\right)\)
. . . . .\(\displaystyle \dfrac{d}{dt}\, y(t)\, =\, y(t)\left(-c\, +\, dx(t)\, -\, fy(t)\right)\)
where \(\displaystyle a\) denotes the growth rate, \(\displaystyle c\) denotes the decay rate, \(\displaystyle b\) and \(\displaystyle d\) denote the coefficients of interspecies conflict, and \(\displaystyle e\) and \(\displaystyle f\) denote the coefficients of intraspecies conflict; \(\displaystyle a,\, b,\, c,\, d,\, e,\) and \(\displaystyle f\) are positive constants.
1) Find all equilibria of the model. In particular, show the sufficient condition for the existence of the coexistence equilibrium.
2) Let \(\displaystyle \left(x^*,\, y^*\right)\) be the coexistence equilibrium. Show that the function
. . . . .\(\displaystyle V (x,\, y)\, =\, dx^* \, \left(\dfrac{x}{x^*}\, -\, 1\, -\, \ln\left(\dfrac{x}{x^*}\right)\right)\, +\, by^*\, \left(\dfrac{y}{y^*}\, -\, 1\, -\, \ln\left(\dfrac{y}{y^*}\right)\right)\)
is the Lyapunov function for the model.
--------------------------------------------------
So, I'm currently on the first part, and I've tried to find the equilibria by setting the derivatives to zero and solving for the variables. I found that
for the derivative of x: x=0 or a-ex-by=0
for the derivative of y: y=0 or -c+dx-fy=0
Which gives us four solutions:
(i) (x,y) = (0,0)
(ii) (x,y) = (0,-c/f)
(iii) (x,y) = (a/e,0)
(iv) (x,y) = ((cb+fa)/(db+fe), (da-ce)/(fe+db))
As I mentioned, the nullcline of y will have a positive incline, not to mention that it crosses the y-axis at a negative value, which is something I haven't seen before. Also, I find it hard to analyze how the derivatives behave depending on the value of the coefficients.
As an example
I think that this should be right: y>a/b => dx/dt = x(a-ex-by) < 0
But when looking at the opposite case, I find it hard to prove that y>a/b => dx/dt = x(a-ex-by) > 0
Where should I go from here?
If it isn't obvious already, I'm completely new to these kind of models (and it has been a while since i handled diff. equations of any kind), so please try to keep it as basic as you can muster
Please help, and please tell me if I need to clarify something.
Thank you in advance!
The assignment is simply to solve a problem concerning the Lotka-Volterra prey predator model, but - to no ones surprise - there is a twist: it includes intraspecies conflict, meaning that there's a couple more coefficients than I've commonly found on webbsites, our textbook or youtube (but I might have missed something obvious). Also, one of the coefficients (c) has a negative sign that gives the nullcline for y a positive incline... The problem is as follows (alternatively, see the attached image):
---------------------------------------------------
Let \(\displaystyle x(t)\) and \(\displaystyle y(t)\) be the populations of prey and predator at time \(\displaystyle t\, \geq\, 0,\) respectively.
Consider the following model:
. . . . .\(\displaystyle \dfrac{d}{dt}\, x(t)\, =\, x(t)\left(a\, -\, ex(t)\, -\, by(t)\right)\)
. . . . .\(\displaystyle \dfrac{d}{dt}\, y(t)\, =\, y(t)\left(-c\, +\, dx(t)\, -\, fy(t)\right)\)
where \(\displaystyle a\) denotes the growth rate, \(\displaystyle c\) denotes the decay rate, \(\displaystyle b\) and \(\displaystyle d\) denote the coefficients of interspecies conflict, and \(\displaystyle e\) and \(\displaystyle f\) denote the coefficients of intraspecies conflict; \(\displaystyle a,\, b,\, c,\, d,\, e,\) and \(\displaystyle f\) are positive constants.
1) Find all equilibria of the model. In particular, show the sufficient condition for the existence of the coexistence equilibrium.
2) Let \(\displaystyle \left(x^*,\, y^*\right)\) be the coexistence equilibrium. Show that the function
. . . . .\(\displaystyle V (x,\, y)\, =\, dx^* \, \left(\dfrac{x}{x^*}\, -\, 1\, -\, \ln\left(\dfrac{x}{x^*}\right)\right)\, +\, by^*\, \left(\dfrac{y}{y^*}\, -\, 1\, -\, \ln\left(\dfrac{y}{y^*}\right)\right)\)
is the Lyapunov function for the model.
--------------------------------------------------
So, I'm currently on the first part, and I've tried to find the equilibria by setting the derivatives to zero and solving for the variables. I found that
for the derivative of x: x=0 or a-ex-by=0
for the derivative of y: y=0 or -c+dx-fy=0
Which gives us four solutions:
(i) (x,y) = (0,0)
(ii) (x,y) = (0,-c/f)
(iii) (x,y) = (a/e,0)
(iv) (x,y) = ((cb+fa)/(db+fe), (da-ce)/(fe+db))
As I mentioned, the nullcline of y will have a positive incline, not to mention that it crosses the y-axis at a negative value, which is something I haven't seen before. Also, I find it hard to analyze how the derivatives behave depending on the value of the coefficients.
As an example
I think that this should be right: y>a/b => dx/dt = x(a-ex-by) < 0
But when looking at the opposite case, I find it hard to prove that y>a/b => dx/dt = x(a-ex-by) > 0
Where should I go from here?
If it isn't obvious already, I'm completely new to these kind of models (and it has been a while since i handled diff. equations of any kind), so please try to keep it as basic as you can muster
Please help, and please tell me if I need to clarify something.
Thank you in advance!
Last edited: