Lotka-Volterra model with two predators

Question

In this, Lotka-Volterra model, we have two predators:

$$\frac{dp}{dt} = ap\left(1-\frac{p}{K}\right) - (b_1q_1+b_2q_2)p$$ $$\frac{dq_1}{dt}=e_1b_1pq_1-m_1q_1$$ $$\frac{dq_2}{dt}=e_2b_2pq_2-m_2q_2.$$

Where $p$ is the prey, $q_1$ is the first predator and $q_2$ is the second predator. Also, $bpq$ is the interaction rate between the species, $m$ is the mortality rate of the predators, $K$ is the carrying capacitance.

We also have that $a=0.2, \ K = 1.7, \ b_1 = 0.1, \ b_2 = 0.2, \ m_1=m_2=0.1, \ e_1 = 1.0,$ and $e_2 =2.0.$ We also, have $p(0) = q_2(0) = 1.7$ and $q_1(0) =1.0$.

I solved this system using Euler's method. I noticed that $\text{predator}_1$ vanishes with time thus two predators cannot co-exist in this model.

My question is, how do I vary the characteristics of the second predator in order to see if there is an equilibrium? Also, what do I vary, in this system, in order to see if there is chaos between the two species?

Answer 1

The coupling product terms $ e_1 b_1, e_2 b_2 $ have values 0.1 and 0.4. So no wonder predator_1 numbers dwindle, itself becoming prey to predator_2. Actually it is a three species model $ ( p, q_1,q_2 ). $

Just reduce $ e_2 b_2 $ values to allow sustain population of predator_1. Confirm by parametrically plotting predator population $ q_1 vs. q_2 $ for several values of $ \dfrac {e_1 b_1} { e_1 b_1 + e_2 b_2}. $

Answer 2

An equilibrium would be a solution of the system of right-hand sides of the equations. To have an equilibrium with both predators present, i.e. $q_1 \ne 0$ and $q_2 \ne 0$, you'll need $e_1 b_1 p = m_1$ and $e_2 b_2 p = m_2$, so $p = \dfrac{m_1}{e_1 b_1} = \dfrac{m_2}{e_2 b_2}$. But then, at this value $p^*$ of $p$ the first equation only specifies a value of $b_1 q_1 + b_2 q_2$, i.e. you won't have a discrete equilibrium point but rather a line segment of equilibrium. Moreover, when this is the case and $p \ne p^*$, $$ \dfrac{q_1'}{q_2'} = \dfrac{e_1 b_1}{e_2 b_2} \dfrac{q_1}{q_2}$$ so $(q_1, q_2)$ varies on a curve $q_1 ^{e_2 b_2} = C q_2^{e_1 b_1}$.