Someone please help! Have spent ages stressing over this and still can't figure out how to do it! need asap.
Question 3. In ecology, the logistic equation is often written in the form
dN divided by dt = rN(1-N/K)
where N = N(t) stands for the size of the population at time t, the constants r
and K stand for the intrinsic growth rate and the carrying capacity of the species,
respectively.
A pond on a �sh farm has carrying capacity of 1000 �sh, intrinsic growth rate
0.3 (when time is measured in months) and is originally stocked with 120 �sh.
(a) Set up a logistic equation for the �sh population N(t) in the pond, with t
measured in months.
(b) Find the size of the population when t = 10.
(c) Can the population reach 1000 at any future time?
even just recommendations on how to begin would be appreciated!
thanks :)
dN/dt = rN(1-N/K)
is a separable equation by rewriting it as:
dN/[rN(1-N/K)] = dt
=>
(K/r)[1/(N(K-N))]dN = dt
split into partial fractions
(1/r)[1/N + 1/(K-N)]dN = dt
Now integrate
(1/r)(ln(N)+ln(K-N))=t + C
Rewriting,
ln(N*(K-N))=r(t+C)
At this point, substitute r and K to find constant C at time t=0.
Hint: C is between 38 and 39.
After that, use the equation to solve for required unknowns.
thank you so much!
You're welcome!
To tackle this problem, let's break it down step by step:
(a) Setting up the logistic equation:
The logistic equation given is:
dN/dt = rN(1-N/K)
We are given:
- Carrying capacity (K) = 1000 fish
- Intrinsic growth rate (r) = 0.3
- Initial population (N) at t = 0 is 120 fish
The logistic equation represents how the population growth rate changes over time as it approaches its carrying capacity.
To set up the logistic equation for the fish population in the pond, we substitute the given values into the equation:
dN/dt = 0.3N(1-N/1000)
(b) Finding the population size at t = 10:
To find the size of the population (N) when t = 10, we will need to solve the differential equation we set up in part (a) using an appropriate method like separation of variables or an integrating factor.
(c) Determining if the population can reach 1000 in the future:
We can determine if the fish population can reach 1000 at any future time by analyzing the behavior of the logistic function. Specifically, we need to observe if the population size (N) grows and eventually stabilizes at the carrying capacity (K = 1000) as time progresses. Solving the differential equation or analyzing its behavior can help us answer this question.
Remember, these steps provide a general guide on how to approach this problem. It is important to carefully analyze the provided information and apply the appropriate mathematical techniques to find the solutions.