Each spring, the deer in Minnesota produce offspring at a...

Each spring, the deer in Minnesota produce offspring at a rate of roughly 1.2 times the total population, while approximately $5 \%$ of the population dies as a result of predators and natural causes. In the fall, hunters are allowed to shoot 3,600 deer. This winter the Department of Natural Resources (DNR) estimates that there are 20,000 deer. Set up an affine iterative equation (9.5) to represent the deer population each subsequent year. Solve the system and find the population in the next 5 years. How many deer in the long term will there be? Using this information, formulate a reasonable policy of how many deer hunting licenses the DNR should allow each fall, assuming one kill per license.

Key Concepts

Difference Equations in Population Modeling

Difference equations are used to model systems that evolve in discrete steps, such as annual changes in a population. These equations help encapsulate the effects of reproduction, mortality, and external factors like harvesting. They provide a framework to project future population sizes based on current demographic processes and interventions over specified time periods.

Policy Formulation from Mathematical Modeling

Mathematical models provide quantitative insights that inform decision-making processes. In wildlife management, for instance, the long-term equilibrium population obtained from an affine iterative model can guide the setting of sustainable hunting quotas. By aligning the number of hunting licenses with the model's predictions, regulatory bodies can balance human interests with ecological sustainability.

Stability Analysis

Stability analysis examines whether the iterative process will converge to an equilibrium or diverge over time. In the context of a linear difference equation, if the effective growth factor is such that the deviations from equilibrium diminish over time, the system is considered stable. This analysis is crucial when applying mathematical models to formulate policies that ensure populations remain sustainable.

Affine Iterative Equations

An affine iterative equation is a recurrence relation in which the next value in a sequence is produced by multiplying the current value by a constant (the growth or decay factor) and then adding or subtracting a constant term. This type of equation, generally expressed as P??? = r·P? + c, captures both proportional changes (e.g., births and deaths) and fixed changes (e.g., sustained removals or inputs) in a discrete time framework.

Equilibrium (Steady State) Analysis

Equilibrium analysis involves finding the fixed point of the iterative process, where the population remains constant from one time step to the next. Mathematically, this occurs when P = r·P + c, allowing analysts to solve for the steady state. This concept is essential for understanding the long-term behavior of a model and assessing sustainability under ongoing processes like growth and harvesting.

Transcript

00:01 First question we're given this scenario that the fish and game department in a certain state is planned to issue hunting permits to control the deer population.

00:08 It is known if the deer population falls below a certain level m, the deer will become extinct.

00:13 It's also known that the deer population rises above the carrying capacity, capital, the population will decrease back to m through disease and malnutrition.

00:23 So for the first one, it says discuss the reasonableness of the following model for the growth rate of the deer population as a function.

00:30 Of time and we're supposed to include a phase line so we're given this function that p prime is equal to r times p and minus p and p minus little m so i want to go ahead and create a phaseline here so here i see that i do have the values where this is going to be zero at p is equal to zero capital m and little m where little m is less than big m and since i'm dealing with population you can have a negative population so my zero is just going to be at the end here and so i see here if i have a value between zero and m for my p i'm going to get that p prime is less than zero here if i have it between little m and big m i'll see that p prime is going to be greater than zero and if i have a bigger than capital m it's going to be less than zero.

01:52 So i see my solutions are going towards zero between these values and for the other two intervals is going to go towards big m.

02:05 And then if you were to take the second derivative here, probably need to use product rule or you could foil all this out to make it a little bit easier.

02:17 You would get an a value and a b value in here.

02:27 Where a is equal to one -third times capital m plus little m minus m squared minus m capital m plus m squared.

02:48 And then b, it's going to be the same thing, but now we're adding.

03:03 And so when you found that derivative and you plug in and see what the sign is between these intervals, you should get less than zero, sorry, greater than zero, less than, greater, less than, and greater.

03:33 So the first question is asking us, how reasonable is this model for the growth rate of the deer population? so looking at this, that means this m right here is unstable.

03:48 So if i have a population that falls below m, i know it's going to go to extinction.

03:54 And i see if i have anything that's greater than little m or greater than big m, then it's just going to fall back to or go towards this level population, which is exactly how this problem explained it in the beginning.

04:12 So i'm going to say, so again, just to write that out, it is reasonable because first, if p is less than m, then p goes to zero as t approaches infinity.

04:34 In other words, extinction.

04:42 If m is less than p, which is less than big m, then p goes to m.

04:50 And same thing if p is greater than m, which is exactly how explained in the problem.

04:58 For part b, we are looking at a different model, so p prime is equal to r times p times m minus p which is basically same thing as this so we're missing this last term here and p minus little m and is asking us to explain how this model differs from the logistic model or how this logistic model differs from the previous model and the difference here is we're not going to have that drop if the population drops below m then p becomes the population population goes to extinct...

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