Population Modelling Homework: Immigration, Growth & Recovery

• missbooty87
In summary, the population model for corals assumes that there is no breeding and constant immigration of juvenile corals, which affects the growth rate of the corals. The equation is first order, nonlinear, and describes the limit to the growth of corals. If there is a "limiting population", the rate of change as M nears it must decrease to 0.
missbooty87

Homework Statement

"The very simple population model for a resource limited population with constant immigration, and no breeding, M'(t) = M(S-M) + I attempts to describe the growth of corals on a reef. Function M(t) represents the biomass of corals."

a - Explain which term gives the immigration of juveniles onto the reef.

b - Describe the presumptions being made about the growth rate of corals at their different ages and sizes.

c - Determine if the biomass of corals tends to a limiting amount as $$t \rightarrow \infty$$ .

d - Suppose a coral reef has completely died, due to excessive cyanide fishing. Find and describe what this model suggests will be the pattern of its recovery.

The Attempt at a Solution

a - Is the immigration denoted as "I", because as immigration is constant, the I has 'constant effects' on the equation?

b - Is it correct to presume that the growth rate of the corals are constant, regardless of their age and size?

c - I can see that its a first order non linear differential equation. But where do I go with this?

d - a regrowth rate represented by a logarithmic function?

The total marks for the 4 questions is 5 marks - if that helps

Thanks to all help received :)

Last edited:
missbooty87 said:

Homework Statement

"The very simple population model for a resource limited population with constant immigration, and no breeding, M'(t) = M(S-M) + I attempts to describe the growth of corals on a reef. Function M(t) represents the biomass of corals."

a - Explain which term gives the immigration of juveniles onto the reef.

b - Describe the presumptions being made about the growth rate of corals at their different ages and sizes.

c - Determine if the biomass of corals tends to a limiting amount as $$t \rightarrow \infty$$ .

d - Suppose a coral reef has completely died, due to excessive cyanide fishing. Find and describe what this model suggests will be the pattern of its recovery.

The Attempt at a Solution

a - Is the immigration denoted as "I", because as immigration is constant, the I has 'constant effects' on the equation?
Yes, I is the only thing that does not depend upon the current mass- it "comes from the outside".

b - Is it correct to presume that the growth rate of the corals are constant, regardless of their age and size?
Yes.

c - I can see that its a first order non linear differential equation. But where do I go with this?
If there is a "limiting population", the rate of change as M nears it must decrease to 0. Is there a value of M such that dM/dt= 0 (so that M is constant)? Set M(S-M) + I = 0 and solve for M.

d - a regrowth rate represented by a logarithmic function?
In other words, solve the problem M'= M(S-M)+ I, M(0)= 0, a separable differential equation. No, M is not logarithmic.

The total marks for the 4 questions is 5 marks - if that helps

Thanks to all help received :)

HallsofIvy said:
If there is a "limiting population", the rate of change as M nears it must decrease to 0. Is there a value of M such that dM/dt= 0 (so that M is constant)? Set M(S-M) + I = 0 and solve for M.

So what I've done is, I've found the value of M using the quadratic formula, this is in terms of S. Is this all I have to do? It feels inadequate...not that I'm undermining your way of thinking... I probably stopped short...

HallsofIvy said:
In other words, solve the problem M'= M(S-M)+ I, M(0)= 0, a separable differential equation.

so if i directly integrate M'= -M^2 + SM + I, which will presumably give the function for M, which is: $$M = \frac{-M^3 }{3} + \frac{SM^2}{2} + IM + C = 0$$ >>> is this what you're talking about? if so, I then substitute M=0? I'm getting lost...

Thank you for your help HallsofIvy :) Much appreciated :)

1. What is population modelling and why is it important?

Population modelling is the use of mathematical and statistical techniques to study and predict changes in the size, structure, and distribution of a population over time. It is important because it helps us understand how populations grow, decline, and recover, and allows us to make informed decisions about resource allocation, policy-making, and other societal issues.

2. How does immigration impact population growth and recovery?

Immigration refers to the movement of people from one country or region to another. It can have a significant impact on population growth and recovery by adding to the total population and potentially increasing the workforce, contributing to economic growth and development. However, it can also lead to challenges such as strain on resources and changes in cultural dynamics.

3. What factors are considered in population modelling?

In population modelling, factors such as birth and death rates, immigration and emigration rates, age and sex distribution, and environmental factors are considered. These factors are used to build mathematical models that can simulate and predict population changes over time.

4. How accurate are population models and predictions?

Population models and predictions are based on available data and assumptions about future trends, so they are not 100% accurate. However, they can provide valuable insights and help us prepare for potential changes in population size and structure. The accuracy of the models also depends on the quality of the data and the assumptions made.

5. How can population modelling be used in policy-making?

Population modelling can inform policy-making by providing insights into future population trends and potential challenges. It can also help evaluate the impact of different policies and interventions on population growth, distribution, and recovery. This can aid in making evidence-based decisions and addressing societal issues related to population dynamics.

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