Suppose a family of salmon fish living off the Alaskan Coast obeys the Malthusian law of population growth $\frac{dp(t)}{dt} = 0.003*p(t)$, where t is measured in minutes. At time t = 0 a group of sharks establishes residence in these waters and begins attacking the salmon. The rate at which salmon are killed by sharks is $0.001p^{2}(t)$, where p(t) is the population of salmon at time t. Moreover, since an undersirable element has moved into their neighborhood, 0.002 salmon per minute leaves the Alaskan waters.

For reference, we will use Salmon that live in the Lake Michigan dataset:

https://docs.google.com/spreadsheets/d/e/2PACX-1vQrV-XqBVdUPO5wzeatk3ncvT4qZiAdEDJJJb9Dc0t9_PpZ_W8Krw1trCpeJJ3S430Ocs0w0PmfL2jP/pubchart?oid=533939390&format=interactive

(a) Modify the Malthusian law to take these factors into account.

According to Wikipedia, Thomas Malthus stated in 1798 the following,

Malthus wrote that all life forms, including humans, have a propensity to exponential population growth when resources are abundant but that actual growth is limited by available resources:

"Through the animal and vegetable kingdoms, nature has scattered the seeds of life abroad with the most profuse and liberal hand. ... The germs of existence contained in this spot of earth, with ample food, and ample room to expand in, would fill millions of worlds in the course of a few thousand years. Necessity, that imperious all pervading law of nature, restrains them within the prescribed bounds. The race of plants, and the race of animals shrink under this great restrictive law. And the race of man cannot, by any efforts of reason, escape from it. Among plants and animals its effects are waste of seed, sickness, and premature death. Among mankind, misery and vice. "

How does the above statement by Mathus relate to the predator-prey population model for sharks and salmon?

First, we must identify and classify each animal as the predator or the prey. In terms of the predator, there is a double edged sword with the shark and salmon, because sharks are an underwater invasive species to marine plants and small marine life, and salmon that eat marine plants to survive. So, there is a fish-plant interaction that may be limited according to Malthus law, and somewhat limited to Verhulst’s Logistic Model of Growth. One may beat out the other to survive on the basis of eating marine plants as a staple food.

(b) Assume that time t = 0 there are 1 million salmon. Find the population p(t). What happens as $t \rightarrow \infin$?

Let’s start with our systems differential equations that we need to help us discover what will happen between the salmon and sharks:

$$ \frac{1}{prey}dx = CarryingCapacity_{prey}*(1 + predator)dt $$

$$ OR $$

$$ \frac{1}{Predator}dx = -CarryingCapacity_{prey}*(1-prey)dt $$

Next, let’s discover the implcit differentation form of the above first order linear differential systems of equations: