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An alternative delayed population growth difference equation model
Sabrina H Streipert1, Gail S K Wolkowicz2
1McMaster University, Hamilton, Ontario, Canada. streipes@mcmaster.ca.
Journal of Mathematical Biology
|August 7, 2021
Summary
This study introduces a new delayed population growth model. Exceeding a critical delay threshold leads to population extinction, while shorter delays result in a stable population size that decreases with increased delay.
Area of Science:
- Population dynamics
- Mathematical biology
- Ecological modeling
Background:
- Traditional population models often assume immediate growth responses.
- Delayed responses in population growth can significantly alter dynamics.
- Existing delayed logistic models do not fully account for mortality during the delay period.
Purpose of the Study:
- To develop an alternative delayed population growth difference equation model.
- To analyze the impact of growth delays on population dynamics.
- To investigate the conditions for population survival and extinction.
Main Methods:
- Modification of the Beverton-Holt recurrence relation.
- Introduction of a delay specifically in the growth contribution.
- Analysis of the delayed difference equation model to identify critical thresholds.
- Application of contraction mapping and componentwise monotone map techniques to prove global asymptotic stability.
Main Results:
- Identification of a critical delay threshold influencing population persistence.
- Prediction of population extinction for delays exceeding the threshold.
- Convergence to a positive, globally asymptotically stable equilibrium for sub-threshold delays.
- Demonstration that equilibrium population size decreases as delay increases.
Conclusions:
- The proposed model provides a more nuanced understanding of delayed population growth.
- Delay duration is a critical factor determining population viability.
- The model predicts a trade-off between delay and population size, with longer delays leading to smaller stable populations or extinction.
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