I am so confused, please help! Pacific salmon populations have discrete breeding

Question

I am so confused, please help! Pacific salmon populations have discrete breeding cycies in which they return from the ocean to streams to reproduce and then die. This occurs every one to five years, depending on the species. (a) Suppose that each fish must first survive predation by bears while swimming upstream, and predation occurs with probability d. After swimming upstream, each fish produces b offspring before dying. The stream is then stocked with m additional newly hatched fish before all fish ten swim out to sea. What is the discrete-time recursion for the population dynamics, assuming that there is no mortaity while at sea? You should count the population immediately before the upstream journey (e) Suppose thet, instead of preying on fish while they swim upstream, bears do so only while the fish are swimming downstream. What is the discrete-time reaursion for the population dynamiks? (Again assume there is no mortality while at sea.)

Explanation / Answer

(a)

Let nt be the number of fish at time point t.

So predation occurs with prob d

Thus remaining fish = nt(1-d)

Then each fish produces b offspring before they die.

Thus remaining fish = nt(1-d) * b - nt(1-d) = nt(1-d) *(b-1)

Then m newly hatched fishes are added.

Thus, remaining fish = nt(1-d)*(b-1) + m

Thus, nt+1 = nt * [(1-d)*(b-1)] + m

(b)

Let nt be the number of fish at time point t.

Here predation doesn't happen upstream.

Then each fish produces b offspring before they die.

Thus remaining fish = nt * b - nt = nt *(b-1)

Then m newly hatched fishes are added.

Thus, remaining fish = nt*(b-1) + m

Now predation occurs with prob d

Thus, nt+1 = [nt *(b-1) + m ] *(1-d)

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