An Alternative Formulation for a Distributed Delayed Logistic Equation.

Area of Science:

• Mathematical Biology
• Ecology
• Population Dynamics

Background:

• Classical logistic delay differential equations (DDEs) can produce unrealistic population oscillations.
• Alternative models, like the discrete delay model by Arino et al., exclude oscillatory behavior.
• Population oscillations are rarely observed in natural populations.

Purpose of the Study:

• To generalize the Arino et al. discrete delay model using a decay-consistent delay.
• To investigate the survival and extinction thresholds of the proposed logistic DDE.
• To explore the evolutionary trends in growth delay using adaptive dynamics.

Main Methods:

• Development of an alternative single species logistic distributed delay differential equation (DDE).
• Establishment of survival and extinction thresholds using Lyapunov functionals and fluctuation lemma.
• Application of adaptive dynamics to determine evolutionary trends in growth delay.

Main Results:

• The generalized model excludes population oscillations, aligning with natural observations.
• A clear threshold for population survival and extinction was established.
• Lyapunov functionals confirmed population convergence to a delay-modified carrying capacity for survival.
• The fluctuation lemma proved population extinction under specific conditions.

Conclusions:

• The study confirms Hutchinson's conjecture regarding evolutionary trends towards shorter mean growth delays.
• The proposed model and findings align with biological evidence of population dynamics.
• The generalized DDE provides a more realistic framework for studying population dynamics with delays.