Dynamics of a Birth-Pulse Single-Species Model with Restricted Toxin Input and Pulse Harvesting

Table 1

Nontrivial equilibria of the two models with birth pulses.

Function	Equilibrium	𝑅0=𝑅^𝑅0 (or 𝑅^𝐵0)
Ricker	𝑥^∗=(1/(1−𝐸))𝑒^𝑑+(𝑘𝑢𝑟/𝑔ℎ)(1−𝑇1)ln𝑅^𝑅0	𝑅^𝑅0=𝑏((1/(1−𝐸))𝑒^𝑑+(𝑘𝑢𝑟/𝑔ℎ)(1−𝑇1)−1)^−1
	𝐶^∗0=(𝑘𝑢/ℎ(𝑔−ℎ))(1−𝑒^−ℎ(1−𝑇1)/1−𝑒^−ℎ)−(𝑘𝑢/𝑔(𝑔−ℎ))(1−𝑒^−𝑔(1−𝑇1)/1−𝑒^−𝑔)
	𝐶^∗𝐸=(𝑢/ℎ)(1−𝑒^−ℎ(1−𝑇1)/1−𝑒^−ℎ)
Beverton-Holt	𝑥^∗=(1/1−𝐸)𝑒^𝑑+(𝑘𝑢𝑟/𝑔ℎ)(1−𝑇1)𝑛𝑞(𝑅^𝐵0−1)	𝑅^𝐵0=(𝑏/𝑞)((1/(1−𝐸))𝑒^𝑑+(𝑘𝑢𝑟/𝑔ℎ)(1−𝑇1)−1)^−1
	𝐶^∗0=(𝑘𝑢/ℎ(𝑔−ℎ))(1−𝑒^−ℎ(1−𝑇1)/1−𝑒^−ℎ)−(𝑘𝑢/𝑔(𝑔−ℎ))(1−𝑒^−𝑔(1−𝑇1)/1−𝑒^−𝑔)
	𝐶^∗𝐸=(𝑢/ℎ)(1−𝑒^−ℎ(1−𝑇1)/1−𝑒^−ℎ)