Population Model Homework: Determine Equilibrium Solutions

In summary, the conversation discussed the simplest useful model for fisher, which is based on the logistic model for population growth and takes into account the harvest proportional to the current population. The model also includes a constant effort, which measures the fraction of the population harvested. The equilibrium solutions were determined for a given effort E, but due to a mistake in factorization, the result was incorrect.
  • #1
tylersmith7690
21
0

Homework Statement



The simplest useful model for fisher comes from the logistic model for population growth, together with a harvest h which is proportional to the current population P, that is,

h=EP,

where the constant E is called the effort. E measures the fraction of the population harvested, so that 0 <= E <= 1. This gives the model

dP/dt = kP(1-(P/a) - EP,

where P(t) is the number of these fish at time t year and k (the natural growth rate) and a( the carrying capacity) are constants for a particular fish polulation. In what follows take k= 1 and a= 4, for simplicity.

a) determine the equilibrium solutions for a given effort E.

my attempt.

dP/dt = P - P^2/4 - EP

dP/dt = ( P + E) (P -1/4)

equilibrium solutions are when dP/dt=0

so P(0)= -E, 1/4.
 
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  • #2
tylersmith7690 said:

Homework Statement



The simplest useful model for fisher comes from the logistic model for population growth, together with a harvest h which is proportional to the current population P, that is,

h=EP,

where the constant E is called the effort. E measures the fraction of the population harvested, so that 0 <= E <= 1. This gives the model

dP/dt = kP(1-(P/a) - EP,

You miss a parentheses: dP/dt = kP(1-(P/a)) - EP

tylersmith7690 said:
where P(t) is the number of these fish at time t year and k (the natural growth rate) and a( the carrying capacity) are constants for a particular fish polulation. In what follows take k= 1 and a= 4, for simplicity.

a) determine the equilibrium solutions for a given effort E.

my attempt.

dP/dt = P - P^2/4 - EP

dP/dt = ( P + E) (P -1/4)

The last equation is wrong.

tylersmith7690 said:
equilibrium solutions are when dP/dt=0

so P(0)= -E, 1/4.

Because of the faulty factorisation, the result is not correct (and physically impossible).

ehild
 

Related to Population Model Homework: Determine Equilibrium Solutions

What is a population model?

A population model is a mathematical representation of the changes in size and composition of a population over time. It takes into account factors such as birth rates, death rates, immigration, and emigration to predict how a population will change.

What is an equilibrium solution in a population model?

An equilibrium solution in a population model is a stable state in which the population size remains constant over time. This occurs when the birth rate equals the death rate, and there is no net migration.

How do you determine equilibrium solutions in a population model?

To determine equilibrium solutions, you must set the population growth equation equal to 0 and solve for the population size. This will give you the population size at which the birth rate equals the death rate and there is no net migration.

Why is it important to understand equilibrium solutions in population modeling?

Understanding equilibrium solutions in population modeling can help predict the long-term growth or decline of a population. It can also inform decision-making for resource management, conservation efforts, and public policy related to population control and sustainability.

What are some limitations of using population models to predict population changes?

Population models are based on assumptions and simplifications, so they may not accurately reflect real-world complexities. They also rely on historical data, which may not always be available or reliable. Additionally, unexpected events or changes in factors such as disease outbreaks or natural disasters can significantly impact the accuracy of population predictions.

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