A discrete two time scales model of a size-structured population of parasitized trees

Rafael Bravo de la Parra; Ezio Venturino; Rafael Bravo de la Parra; Ezio Venturino

doi:10.3934/mbe.2024309

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 9: 7040-7066. doi: 10.3934/mbe.2024309

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A discrete two time scales model of a size-structured population of parasitized trees

Rafael Bravo de la Parra ^{1
,
,},
Ezio Venturino ²

1.
U.D. Matemáticas, Universidad de Alcalá, Alcalá de Henares, Spain
2.
Dipartimento di Matematica "Giuseppe Peano", Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy; Laboratoire Chrono-Environnement, Université de Franche-Comté, 16 route de Gray, Besançon, 25030, France Member of the INdAM research group GNCS

Received: 24 June 2024 Revised: 12 August 2024 Accepted: 19 August 2024 Published: 09 September 2024

The work presented a general discrete-time model of a population of trees affected by a parasite. The tree population was considered size-structured, and the parasite was represented by a single scalar variable. Parasite dynamics were assumed to act on a faster timescale than tree dynamics. The model was studied based on an associated nonlinear matrix model, in which the presence of the parasites was only reflected in the value of its parameters. For the model in all its generality, an explicit condition of viability/extinction of the parasite/tree community was found. In a simplified model with two size-classes of trees and particular forms of the vital rates, it was shown that the model undergoes a transcritical bifurcation and, likewise, a period-doubling bifurcation. It was found that, for any tree fertility rate that makes them viable without a parasite, if the parasite sufficiently reduces the survival of young trees, it can lead to the extinction of the entire community. The same cannot be assured if the parasite acts on adult trees. In situations where a high fertility rate coupled with a low survival rate of adult trees causes a non-parasitized population of trees to fluctuate, a parasite sufficiently damaging only young trees can stabilize the population. If, instead, the parasite acts on adult trees, we can find a destabilization condition on the tree population that brings them from a stable to an oscillating regime.
- discrete-time model,
- size-structured population,
- time scales,
- matrix population model,
- period-doubling bifurcation
Citation: Rafael Bravo de la Parra, Ezio Venturino. A discrete two time scales model of a size-structured population of parasitized trees[J]. Mathematical Biosciences and Engineering, 2024, 21(9): 7040-7066. doi: 10.3934/mbe.2024309

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Abstract

The work presented a general discrete-time model of a population of trees affected by a parasite. The tree population was considered size-structured, and the parasite was represented by a single scalar variable. Parasite dynamics were assumed to act on a faster timescale than tree dynamics. The model was studied based on an associated nonlinear matrix model, in which the presence of the parasites was only reflected in the value of its parameters. For the model in all its generality, an explicit condition of viability/extinction of the parasite/tree community was found. In a simplified model with two size-classes of trees and particular forms of the vital rates, it was shown that the model undergoes a transcritical bifurcation and, likewise, a period-doubling bifurcation. It was found that, for any tree fertility rate that makes them viable without a parasite, if the parasite sufficiently reduces the survival of young trees, it can lead to the extinction of the entire community. The same cannot be assured if the parasite acts on adult trees. In situations where a high fertility rate coupled with a low survival rate of adult trees causes a non-parasitized population of trees to fluctuate, a parasite sufficiently damaging only young trees can stabilize the population. If, instead, the parasite acts on adult trees, we can find a destabilization condition on the tree population that brings them from a stable to an oscillating regime.

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