Analytical and numerical bifurcation analysis of a forest ecosystem model with human interaction

¹ Dipartimento di Agraria, Universitá degli Studi di Napoli Federico II, Reggia di Portici - Via Universitá, Portici 100 – 80055, Italy
² Mathematical Institute, Rostock University, Ulmenstrasse 69, Rostock D-18057, Germany
³ Istituto di Scienze e Tecnologie per l’Energia e la Mobilità Sostenibile, Consiglio Nazionale delle Ricerche, Piazzale Tecchio, Napoli 80 – 80125, Italy
⁴ Dipartimento di Agraria, Universitá degli Studi di Napoli Federico II, Reggia di Portici - Via Universitá, Portici 100 – 80055, Italy
⁵ Dipartimento di Matematica e Applicazioni “Renato Caccioppoli’’, Universitá degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, Napoli I-80126, Italy

^* Corresponding author: constantinos.siettos@unina.it

Received: 25 October 2019
Accepted: 28 July 2020

Abstract

We perform both analytical and numerical bifurcation analysis of an alternating forest and grassland ecosystem model coupled with human interaction. The model consists of two nonlinear ordinary differential equations incorporating the human perception of the value of the forest. The system displays multiple steady states corresponding to different forest densities as well as regimes characterized by both stable and unstable limit cycles. We derive analytically the conditions with respect to the model parameters that give rise to various types of codimension-one criticalities such as transcritical, saddle-node, and Andronov–Hopf bifurcations and codimension-two criticalities such as cusp and Bogdanov–Takens bifurcations at which homoclinic orbits occur. We also perform a numerical continuation of the branches of limit cycles. By doing so, we reveal turning points of limit cycles marking the appearance/disappearance of sustained oscillations. Such critical points that cannot be detected analytically give rise to the abrupt loss of the sustained oscillations, thus leading to another mechanism of catastrophic shifts.