The Allee effect For the model of population dynamics from...

The Allee effect For the model of population dynamics
from Example $3,$ use the local stability criterion to verify that
$\hat{N}=0$ and $\hat{N}=K$ are locally stable whereas $\hat{N}=a$ is
unstable.

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Key Concepts

Equilibrium Points

Equilibrium points in population dynamics are the population levels at which the system remains constant over time because the birth, death, immigration, and emigration rates balance out. These points are found by setting the rate of change (or the recurrence relation) equal to zero and solving, indicating where the population has no net growth or decline.

Local Stability Criterion

The local stability criterion is a method used to determine the behavior of a dynamical system near its equilibrium points. By examining the sign and magnitude of the derivative of the population function at an equilibrium point, one can infer whether small perturbations will decay (indicating stability) or grow (indicating instability).

Allee Effect

The Allee effect describes a phenomenon in population biology where a population has a critical threshold below which it tends to decline toward extinction. This effect is particularly significant because it suggests that populations benefits from a minimum density, and below this level, the per capita growth rate deteriorates, leading to instability of equilibrium points at low population sizes.

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