A predator-prey model Consider the predator-prey model
x...

Question

A predator-prey model Consider the predator-prey model

x′(t) = −4x + 2xy, y′(t) = 5y − xy

c. Find the equilibrium points for the system.

Accepted Answer

Welcome back, everyone. Given the system, P T equals -2P plus 5PQ and Q T equals 8Q minus 4 PQ, find all equilibrium points. For this problem, let's recall that we identify our equilibrium points by setting the first derivative equal to 0. Let's begin with our first equation P of T equals 0. It essentially means that negative 2p. Plus 5 PQ equals 0. We can solve it by factoring out. P. in we're going to have 2 minus 5 Q. And our product is equal to 0. Solving this equation, we can show that either P equals 0 or 2 minus 5 Q equals 0. So we have one possible equilibrium point B equals 0, and now for the second part, we are going to rearrange it and show that 2 equals 5Q or Q equals 2 divided by 5, right? And now let's focus on the second equation. We have Q of T, we're going to set it equals to 0 to identify additional potential equilibrium points. 8Q minus 4PQ is equal to 0. In this case we can factor out 4 Q. In C we have 2 minus p. Is equal to Cyril. Solving this equation, we have Q equals 0 because we have 4 Q as our first factor. Or 2 minus p is equal to 0. Meaning P is equal to 2 for the latter part. So now we have identified two P values, P equals 0 and P equals 2. And 2 Q values Q equals 0 and Q equals 2 divided by 5. We want to identify combinations of P and Q. That produce both derivatives equal to 0, right? So now notice that if we take P equals 0. Than you. Is going to be 0 for the first equation. It satisfies the first equation, right? Because if P is equal to 0, Q can be any value. So if P is equal to 0, Q can be any value. But we also want to simultaneously satisfy the second equation, which is Q of T equals 8 q minus 4 PQ, right? So if P is equal to 0, then we get Q of T equals 8Q. So Q can only be 0 if P is equal to 0, right? Meaning the only combinations of P and Q for which P T equals Q T. Equals 0 would be 0.0, meaning P is 0 and Q is 0. Or let's check any other combinations. If P is equal to 2, right? We are going to have our second equation equal to 0, and our first equation would be -4 plus, let's write it down, -4 + 5 multiplied by 2 is 10-Q. So we want to ensure that this is equal to 0, right? We can now show that 10-Q is equal to 4, or basically Q is equal to 25th. So our second possibility is to have. P of 2 and Q of 2 divided by 5, so we have two possible combinations 0.0 or 2. 2 divided by 5, we can also say and, right, because we're looking at all the possible equilibrium points instead of or. Well then we have our final answer and thank you for watching.

A predator-prey model Consider the predator-prey model
x′(t) = −4x + 2xy, y′(t) = 5y − xy
c. Find the equilibrium points for the system.

Key Concepts

Equilibrium Points in Dynamical Systems

Solving Systems of Nonlinear Equations

Predator-Prey Model Dynamics

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