Oscillations in marine and insect populations with delayed density-dependence
1Physique des Systèmes Dynamiques, Université Libre de Bruxelles, Campus Plaine, C.P. 231, 1050 Bruxelles, Belgium.
Mathematical Biosciences and Engineering : MBE
|May 14, 2026
Summary
This study models population dynamics using a delay differential equation, revealing that Hopf bifurcations can cause sustained oscillations in invertebrate and insect populations with limited resources. Analytical and numerical methods confirm these population dynamics.
Area of Science:
- Ecology
- Mathematical Biology
- Dynamical Systems
Background:
- Limited resources often lead to density-dependent population regulation.
- Delay differential equations are suitable for modeling populations with time lags in their life cycles.
- Hopf bifurcations can generate oscillations in biological systems.
Purpose of the Study:
- To model the limited growth of sessile marine invertebrates and insect populations using a scalar delay differential equation with threshold nonlinearity.
- To investigate the conditions under which sustained oscillations arise due to Hopf bifurcations.
- To analyze the properties of time-periodic solutions and their bifurcations.
Main Methods:
- Formulation of a scalar delay differential equation with threshold nonlinearity.
- Application of the method of steps to construct time-periodic solutions.
- Analysis of bifurcation properties.
- Comparison of analytical results with numerical simulations.
Main Results:
- Sustained oscillations are possible depending on juvenile settlement and mortality rates.
- Analytical findings align with numerical simulations of the delay differential equation.
- Hopf bifurcation points were analyzed, especially under high juvenile settlement rates.
- Low adult mortality rates lead to significant changes in the periodic waveform.
Conclusions:
- The developed delay differential equation accurately describes population dynamics with limited growth and resources.
- Hopf bifurcations are a key mechanism for generating population oscillations in these models.
- The study provides insights into how settlement and mortality rates influence population stability and waveform characteristics.
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