# American Institute of Mathematical Sciences

• Previous Article
Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps
• DCDS-B Home
• This Issue
• Next Article
Assignability of dichotomy spectra for discrete time-varying linear control systems

## 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

Received  December 2018 Revised  April 2019 Published  September 2019

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

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.

Citation: Elena Travaglia, Valentina La Morgia, Ezio Venturino. Poxvirus, red and grey squirrel dynamics: Is the recovery of a common predator affecting system equilibria? Insights from a predator-prey ecoepidemic model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019200
##### References:

show all references

##### References:
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 $G$, $R$, $I$, $M$
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 $G$, $R$, $I$, $M$
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 $G$, $R$, $I$, $M$
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 $G$, $R$, $I$, $M$
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 $G$, $R$, $I$, $M$
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 $G$, $R$, $I$, $M$
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 $G$, $R$, $I$, $M$
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 $G$, $R$, $I$, $M$
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
 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
 [1] Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187 [2] Sebastian J. Schreiber. On persistence and extinction for randomly perturbed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 457-463. doi: 10.3934/dcdsb.2007.7.457 [3] Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807 [4] Liang Kong, Tung Nguyen, Wenxian Shen. Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1613-1636. doi: 10.3934/cpaa.2019077 [5] El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449 [6] Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355 [7] Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447 [8] John Erik Fornæss. Sustainable dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361 [9] Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935 [10] Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91 [11] Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1 [12] Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432 [13] Josiney A. Souza, Tiago A. Pacifico, Hélio V. M. Tozatti. A note on parallelizable dynamical systems. Electronic Research Announcements, 2017, 24: 64-67. doi: 10.3934/era.2017.24.007 [14] Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116 [15] Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1677-1692. doi: 10.3934/dcds.2016.36.1677 [16] Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531 [17] Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587 [18] Alexander Sakhnovich. Dynamical canonical systems and their explicit solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1679-1689. doi: 10.3934/dcds.2017069 [19] Jérôme Rousseau, Paulo Varandas, Yun Zhao. Entropy formulas for dynamical systems with mistakes. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4391-4407. doi: 10.3934/dcds.2012.32.4391 [20] Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215

2018 Impact Factor: 1.008