# Mathematical Model: Discrete Logistic Growth and Fish Harvesting

Please read the following scenario, modify the given mathematical model as instructed, and present your model and analysis. Please respond using difference equations (not differential equations).

Non-Linear Discrete Dynamical Logistic Growth Model with Harvesting

Scenario
The Ministry of Fisheries has been tracking the coastal salmon fishery and is considering reopening a currently protected population of salmon to harvesting. The population was previously harvested to the brink of collapse, the fishery was closed, and the population has now recovered. Not wanting to repeat its past failure to properly manage the fishery, the Ministry is seeking to develop and create a model of the fishery with harvesting. Existing data on the salmon population was used to create a discrete logistics model that accurately describes the growth of the population as
$N_{t+1}=N_t+RN_t(1-N_t/K)$
Where $N_t$ denotes the number of salmon in generation $t$, $R$ is the intrinsic rate of growth and $K$ is the carrying capacity

Your task is to adapt the above model to investigate how harvesting will impact the salmon population. Two different management strategies have been proposed.
- The first strategy is to allow a fixed number of salmon, h, to be harvested each year before the salmon reproduce.
- The second strategy is to allow a fixed proportion of the salmon population, H , to be harvested each year before the salmon reproduce.

Modify the discrete logistic model given above to investigate each of these scenarios. Provide a full analysis of each scenario and a recommendation as to the better management strategy and the level of harvesting that should be allowed.

Note that you may find it easier if you rewrite the above equation in the form $N_{n+1}=rN_t(1-N_t)$, introducing the new parameter $r$, defined as $r=R/K$

Please be sure to provide a full and thorough analysis of the mathematical model. Thank you in advance for your help.

• Nirenberg

What do you mean by "full and through analysis". You want those adapted equations to be solved?

• Nirenberg

Or you want analysis of the critical points and their stability, and long term behavior of the solutions?

• Euclid

An analysis of the two scenarios and which one is better. Please include all calculations and if you could comment on the equilibrium, stability etc., if related to the model. Maybe have some examples with different values to explain the behaviour.

• Euclid

Thank you

• Nirenberg

The offered bounty is too low for all that work. This could easily take over an hour to answer.

• Nirenberg

If you want to know what a fair bounty should be, just imagine how much a tutor would charge to carefully work it out.

• Euclid

I have increased the bounty as much as I could afford and would really appreciate a thorough response. Thank you :)

• Mathe

Would an analysis of the equilibrium points and its stability suffice?

• Euclid

Hi Rage, I am also looking for the analysis of which scenario is recommended and the level of harvesting allowed.

• Mathe

Yes, I can make a recommendation based on the stability of the equilibrium points. Would that suffice?

• Euclid

Yes, so long as its well explained. I need to know what you do with the equation to analyze each scenario. Thank you!