# How do I find the analytical solutions to Lotka Volterra equations?

The Lotka Volterra equations are used to calculate the rate of change of predator, $y$, and prey, $x$, populations over time ($t$),and are shown below:

$$\frac{dx}{dt}=ax-bxy$$

$$\frac{dy}{dt}=-cy+dxy$$

where $a$, $b$, $c$ and $d$ are parameters, and $x$ and $y$ are greater than zero.

I have been looking at online methods, used by mathematicians, however, researchers seem to be solving the parameters through computer softwares like Matlab and Maple. I could not find analytic i.e algebraic solutions to the parameters. My question is,can the parameters be determined algebraically? As a high school student, my school does not allow me to use high tech computer softwares like maple, and I have been to told find the parameters using algebra. However, online research and mathematical literature seems to suggest, that the only way to determine the parameters is through a computer analyses, and not with algebraic rigour.

It would be great, if I could get some sort of a direction, as to what I should do to figure out what the parameters are through algebra, if the population of the prey oscillates between 200 and 400 sheep and the predator population oscillates between 50 and 100.

• Are the equations written correctly? Did you mean $a x$?
– Moo
Jan 16 '18 at 17:35
• @Moo Hey, thanks! It was a mistake. I fixed it :) Jan 16 '18 at 17:41
• You can't find $x(t)$ and $y(t)$ explicitly, but you can find a constant of motion $V(x,y) = \text{constant}$. See here, for example: en.wikipedia.org/wiki/… Jan 16 '18 at 17:55
• @Hans Lundwark Thankyou! math.stackexchange.com/questions/1853612/… What do you think of the method presented here as an answer for the calculation of the parameters.? Jan 16 '18 at 18:03
• @HansLundmark I have already derived the linear function for the Lotka Volterra equations in terms of V(x,y)=constant; V (x,y)=constant. But the problem is still there, is there a method for calculating the parameters algebraically? Jan 16 '18 at 18:05