In order to model the interaction of two species with continuous time, we need a system of two differential equations. Typically, the system will look like: dx = f x, y = terms describing growth of x + interaction terms dt dy

= g x, y = terms describing growth of y + interaction terms dt For example: dx = K2 x C 0 .5 x$y dt dy

= 3 y K x$y dt Question 8 What would happen to each species in the absence of the other? How does the presence of each species affect the other?

Question 9 Find the steady states of the system. [ The pair (x*, y*) is a steady state if and only if both f x*, y*) = 0 and g x*, y*) = 0]

We will now plot the solution trajectories x t , y t solutions. in

the xy plane and interpret the behavior of

Week 5 -- September 26

The Community Matrix

One way to assess the effect of each species upon the other species in the system is to examine the matrix J=

vf vx vf vy vg vx vg vy The signs of the partial derivatives indicate whether an increase in a given species increases or decreases the growth rates. vf O0 . vy This means that an increase in species Y has causes an ___________________ in the growth rate of species X.

For example, suppose

Similarly

vg indicates whether an increase in X is beneficial or detrimental to species Y. vx These are the "inter-specific" terms -- meaning they express ineractions between different species.

"Intra-specific" refers to interactions between members of the same species.

The first Example (henceforth called Example P ) can represent a Predator-Prey interaction.

Question 10 How do the solution trajectories appear to behave? Is there an asymptotically stable steady state?

Discuss whether/to what extent this model is reasonable.

Question 11 What type of interaction could the following systems represent?

(C1)

dx

= 2 x K x2 K x$y

2

dt dy = 4 y Ky2 K3 x$y dt dx

= 3 x K x2 K 2 x$y

1

dt dy = 4 y K3 y2 K1 x$y dt Question 12 How does each species behave in the absence of the other?

(C2)

Question 13 Find the steady states.

Question 14 Based on the solution trajectories, what happens in the long run for these systems? Can competing species coexist?

STABILITY

How does one determine stability of the SS for the various