In this paper, we study an $ n $-species population model with periodically varying biotic interactions that can continuously span mutualism, competition, and consumer–resource outcomes, as a natural extension of a previous two-species framework. Each pairwise interaction is modulated by a periodic function representing the time-dependent fraction of events with positive effects, thus capturing ecologically realistic scenarios where the sign and strength of interactions fluctuate seasonally. Under mild assumptions on the model parameters, we prove that there exists an explicit threshold for the intraspecific competition coefficients above which the system admits a unique positive $ T $-periodic coexistence state that is asymptotically stable. Some illustrative numerical examples are included. The results support the ecological principle that sufficiently strong intraspecific regulation is the key mechanism ensuring long-term coexistence in communities with temporally variable interaction outcomes.
Citation: Ruth I. Oliva-Zúniga, José M. Gómez, Pedro J. Torres. A periodic model for biotic interactions of many species with a continuous spectrum of outcomes[J]. Mathematical Biosciences and Engineering, 2026, 23(9): 2613-2633. doi: 10.3934/mbe.2026094
In this paper, we study an $ n $-species population model with periodically varying biotic interactions that can continuously span mutualism, competition, and consumer–resource outcomes, as a natural extension of a previous two-species framework. Each pairwise interaction is modulated by a periodic function representing the time-dependent fraction of events with positive effects, thus capturing ecologically realistic scenarios where the sign and strength of interactions fluctuate seasonally. Under mild assumptions on the model parameters, we prove that there exists an explicit threshold for the intraspecific competition coefficients above which the system admits a unique positive $ T $-periodic coexistence state that is asymptotically stable. Some illustrative numerical examples are included. The results support the ecological principle that sufficiently strong intraspecific regulation is the key mechanism ensuring long-term coexistence in communities with temporally variable interaction outcomes.
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Ruth I. Oliva-Zúniga, José M. Gómez, Pedro J. Torres. A periodic model for biotic interactions of many species with a continuous spectrum of outcomes[J]. Mathematical Biosciences and Engineering, 2026, 23(9): 2613-2633. doi: 10.3934/mbe.2026094
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