Oscillations in marine and insect populations with delayed density-dependence

Thomas Erneux; Yang Kuang; Thomas Erneux; Yang Kuang

doi:10.3934/mbe.2026049

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 5: 1340-1355. doi: 10.3934/mbe.2026049

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Oscillations in marine and insect populations with delayed density-dependence

Thomas Erneux ^{1
,
,},
Yang Kuang ^2,†

1.
Physique des Systèmes Dynamiques, Université Libre de Bruxelles, Campus Plaine, C.P. 231, 1050 Bruxelles, Belgium
2.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
^†In memory of Stephen Gourley who was our friend, colleague, and a gifted teacher.

Received: 10 February 2026 Revised: 16 March 2026 Accepted: 20 March 2026 Published: 02 April 2026

A scalar delay differential equation with threshold nonlinearity has been formulated to describe the limited growth of sessile marine invertebrates and insect populations when resources are limited. Depending on the settlement and mortality rates of juveniles, sustained oscillations generated by Hopf bifurcations are possible. We construct these time-periodic solutions by using the method of steps and analyze their bifurcation properties. All our analytical results are in agreement with those obtained from numerical simulations of the original delay differential equation. Particular attention is devoted to the Hopf bifurcation points in the limit of a high juvenile settlement rate. In addition, we identify important changes in the periodic waveform when adults exhibit low mortality rates.
- oscillation,
- periodic solution,
- delay differential equation,
- marine population model,
- density-dependence
Citation: Thomas Erneux, Yang Kuang. Oscillations in marine and insect populations with delayed density-dependence[J]. Mathematical Biosciences and Engineering, 2026, 23(5): 1340-1355. doi: 10.3934/mbe.2026049

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Abstract

A scalar delay differential equation with threshold nonlinearity has been formulated to describe the limited growth of sessile marine invertebrates and insect populations when resources are limited. Depending on the settlement and mortality rates of juveniles, sustained oscillations generated by Hopf bifurcations are possible. We construct these time-periodic solutions by using the method of steps and analyze their bifurcation properties. All our analytical results are in agreement with those obtained from numerical simulations of the original delay differential equation. Particular attention is devoted to the Hopf bifurcation points in the limit of a high juvenile settlement rate. In addition, we identify important changes in the periodic waveform when adults exhibit low mortality rates.

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