Poxvirus, red and grey squirrel dynamics: Is the recovery of a common predator affecting system equilibria? Insights from a predator-prey ecoepidemic model

Elena Travaglia; Valentina La Morgia; Ezio Venturino

doi:10.3934/dcdsb.2019200

Article Contents

2020, Volume 25, Issue 6: 2023-2040. Doi: 10.3934/dcdsb.2019200

This issue Previous Article Preface: Population dynamics in epidemiology and ecology Next Article Takens–Bogdanov singularity for age structured models

Poxvirus, red and grey squirrel dynamics: Is the recovery of a common predator affecting system equilibria? Insights from a predator-prey ecoepidemic model

1.
Dipartmento di Matematica "Giuseppe Peano", Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy
2.
Institute for Environmental Protection and Research (ISPRA), via Ca' Fornacetta 9, 40064, Ozzano Emilia (BO), Italy
3.
Dipartmento di Matematica "Giuseppe Peano", Università di Torino, via Carlo Alberto 10

^* Corresponding author: Ezio Venturino
^* Corresponding author: Ezio Venturino

Received: November 30, 2018

Revised: March 31, 2019

Early access: September 2019

Published: June 2020

The third author is member of the INdAM research group GNCS.

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Abstract

In Europe, the Eastern grey squirrel is an allochthonous species causing severe impacts on the native red squirrel. The invasive species establishes complex relationships with the native one, out-competing it through resource and disease-mediated competition. However, recent research shed light on the potential role of a predator, the pine marten, in reversing the outcome of the competition between squirrels. Here, we investigate this hypothesis developing a one predator-two prey ecoepidemic model, including disease (squirrel poxvirus) transmission. We assess the equilibria of the dynamical system and investigate their sensitivity to ecosystem parameters changes through numerical simulations.

Our analysis reveals that the system is more likely to evolve toward points where the red squirrels thrive than toward a disease-and-red-squirrels-free point. Although the disease is likely to remain endemic in the system, the introduction of the pine marten destabilizes previous equilibria, favouring the native squirrel and facilitating wildlife managers in their efforts to protect it. Nevertheless, the complete eradication of grey squirrels could be achieved only for specified values of the predation rates and pine marten carrying capacity. The active management of grey squirrel populations remains therefore necessary to try to eradicate the invader from the system.

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Mathematics Subject Classification: Primary: 92D40, 92D25; Secondary: 92D30.

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Figure 1. Equilibria behaviour in the $ u-h $ parameter space. The height of the surface represents the value of each population at equilibrium; clockwise from top left $ G $, $ R $, $ I $, $ M $

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Figure 2. Equilibria behaviour in the $ h-r $ parameter space. The height of the surface represents the value of each population at equilibrium; clockwise from top left $ G $, $ R $, $ I $, $ M $

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Figure 3. Equilibria behaviour in the $ h-b $ parameter space. The height of the surface represents the value of each population at equilibrium; clockwise from top left $ G $, $ R $, $ I $, $ M $

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Figure 4. Equilibria behaviour in the $ h-\mu $ parameter space. The height of the surface represents the value of each population at equilibrium; clockwise from top left $ G $, $ R $, $ I $, $ M $

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Figure 5. Equilibria behaviour in the $ \mu-r $ parameter space. The height of the surface represents the value of each population at equilibrium; clockwise from top left $ G $, $ R $, $ I $, $ M $

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Figure 6. Equilibria behaviour in the $ u-c $ parameter space. The height of the surface represents the value of each population at equilibrium; clockwise from top left $ G $, $ R $, $ I $, $ M $

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Figure 7. Equilibria behaviour in the $ u-\mu $ parameter space. The height of the surface represents the value of each population at equilibrium; clockwise from top left $ G $, $ R $, $ I $, $ M $

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Figure 8. Equilibria behaviour in the $ u-s $ parameter space. The height of the surface represents the value of each population at equilibrium; clockwise from top left $ G $, $ R $, $ I $, $ M $

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Table 1. Possible ecosystem's equilibria other than coexistence

Equilibria	Feasibility	Stability
$ E_0 =\left( 0, 0, 0, 0 \right) $	unconditional	unstable
$ E_1 =\left( 0, 0, 0, M_1, \right) $	unconditional	(3)
$ E_2 =\left( G_2, 0, 0, 0 \right) $	unconditional	unstable
$ E_3=\left(G_3, 0, 0, M_3\right) $	(4)	(5)
$ E_4 =\left( 0, R_4, 0, 0 \right) $	unconditional	unstable
$ E_5 =\left( 0, R_5, I_5, 0 \right) $	(6)	unstable
$ E_6=\left(0, R_6, 0, M_6\right) $	(7)	(8)
$ E_7=\left(0, R_7, I_7, M_7\right) $	(9), (10), (11)	(10), (12)
$ E_8=\left( G_8, R_8, I_8, 0 \right) $	conditional	unstable

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