# American Institute of Mathematical Sciences

June  2020, 25(6): 2023-2040. doi: 10.3934/dcdsb.2019200

## 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  June 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (6) : 2023-2040. doi: 10.3934/dcdsb.2019200
##### References:
 [1] F. Barbara, V. La Morgia, V. Parodi, G. Toscano and E. Venturino Analysis of the incidence of poxvirus on the dynamics between red and grey squirrels, Mathematics, 6 (2018), 113. doi: 10.3390/math6070113. [2] S. Bertolino, Attivazione di un Progetto di Monitoraggio Estansivo ed Intensivo Dello Scoiattolo Grigio (Sciurus Carolinensis) in Piemonte, Technical report, Regione Piemonte, 2004. [3] S. Bertolino and P. Genovesi, The application of the European strategy on invasive alien species: An example with introduced squirrels, Hystrix, the Italian Journal of Mammalogy, 16 (2005), 59-69. [4] S. Bertolino, A. Martinoli and L. A Wauters, Risk Assessment for Sciurus carolinensis (grey squirrel), In H. Roy, editor, Invasive Alien Species-Framework for the Identification of Invasive Species of EU Concern, Natural Environment Research Council, (2014), 298. [5] L. Boitani, S. Lovari and A. Vigna Taglianti, Fauna d'Italia. Mammalia III: Carnivora-Artiodactyla, Edizioni Calderini de Il Sole-24 Ore, Bologna, 2003. [6] F. Cassola, Sciurus carolinensis, eastern gray squirrel, The IUCN Red List of Threatened Species, (2018), 1–8. [7] J. Chantrey, T. D. Dale, J. M. Read, S. White, F. Whitfield, D. Jones, C. J. McInnes and M. Begon, European red squirrel population dynamics driven by squirrelpox at a gray squirrel invasion interface, Ecology and Evolution, 4 (2014), 3788-3799.  doi: 10.1002/ece3.1216. [8] John Gurnell, The effects of food availability and winter weather on the dynamics of a grey squirrel population in Southern England, Journal of Applied Ecology, 33 (1996), 325-338.  doi: 10.2307/2404754. [9] J.Gurnell, The Natural History of Squirrels, Helm, London. [10] J. Gurnell, Squirrel numbers and the abundance of tree seeds, Mammal Review, 13 (1983), 133-148.  doi: 10.1111/j.1365-2907.1983.tb00274.x. [11] J. Gurnell, L. A. Wauter, P. W. W. Lurz and G. Tosi, Alien species and interspecific competition: Effects of introduced eastern grey squirrels on red squirrel population dynamics, Journal of Animal Ecology, 73 (2004), 26-35.  doi: 10.1111/j.1365-2656.2004.00791.x. [12] S. Harris, P. Morris, S. Wray and D. Yalden, A Review of British Mammals: Population Estimates and Conservation Status of British Mammals Other than Cetaceans, Technical report, Joint Nature Conservation Committee, Peterborough, 1995. [13] S. Harris, C. D. Soulsbury and G. Iossa, Is Culling Grey Squirrels a Viable Tactic to Conserve Red Squirrel Populations?, Technical report, Advocates for animals, 2006. [14] R. E. Kenward and J. M. Tonkin, Red and grey squirrels: Some behavioural and biometric differences, Journal of Zoology, 209 (1986), 279-281.  doi: 10.1111/j.1469-7998.1986.tb03583.x. [15] P. W. W. Lurz, S. P. Rushton, L. A. Wauters, S. Bertolino, I. Currado, P. Mazzoglio and M. D. F. Shirley, Predicting grey squirrel expansion in North Italy: A spatially explicit modelling approach, Landscape Ecology, 16 (2001), 407-420. [16] J. L. MacPherson, H. Denman, D. Tosh, C. McNicol and E. C. Halliwell, A review of the current evidence for impacts of the pine marten (Martes martes) on non-native and native squirrel populations, In The Grey Squirrel Ecology and Managamente of an Invasive Species in Europe, ESI, European Squirrel Initiative, 289–304. [17] A. Martinoli, S. Bertolino, D. G. Preatoni, A. Balduzzi, A. Marsan, P. Genovesi, G. Tosi and L. A. Wauters, Headcount 2010: The multiplication of the grey squirrel populations introduced to Italy, Hystrix, the Italian Journal of Mammalogy, 21 (2010), 127-136. [18] C. J. McInnes, A. R. Wood, K. Thomas, A. W. Sainsbury, J. Gurnell, F. J. Dein and P. F. Nettleton, Genomic characterization of a novel poxvirus contributing to the decline of the red squirrel (Sciurus vulgaris) in the UK, Journal of General Virology, 87, 2115–2125. [19] A. D. Middleton, 38. The Ecology of the American Grey Squirrel (Sciurus carolinensis Gmelin) in. the British Isles, Proceedings of the Zoological Society of London, 100 (1930), 809-843. [20] S. D. Montgomery, J. B. Whelan and H. S. Mosby, Bioenergetics of a woodlot gray squirrel population, The Journal of Wildlife Management, 39 (1975), 709-717.  doi: 10.2307/3800232. [21] H. S. Mosby, The influence of hunting on the population dynamics of a woodlot gray squirrel population, The Journal of Wildlife Management, 33 (1969), 59-73.  doi: 10.2307/3799650. [22] A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in Britain, Proceedings of the Royal Society of London. Series B, Biological sciences, 238 (1989), 113-125.  doi: 10.1098/rspb.1989.0070. [23] J. C. Reynolds, Details of the geographic replacement of the red squirrel (Sciurus vulgaris) by the grey squirrel (Sciurus carolinensis) in eastern england, The Journal of Animal Ecology, 54 (1985), 149–162. [24] C. Romeo, L. A. Wauters and N. Ferrari, 11. Parasite of grey squirrels: An additional threat to red squirrels in Italy?, In The Grey Squirrel Ecology and Managamente of an Invasive Species in Europe, ESI, European Squirrel Initiative, (2016), 193–209. [25] J. J. Rowe, Squirrel management, Mammal Review, 13 (1983), 173-181.  doi: 10.1111/j.1365-2907.1983.tb00279.x. [26] S. P. Rushton, P. W. W. Lurz, J. Gurnell, R. Fuller, Modelling the spatial dynamics of parapoxvirus disease in red and grey squirrels: A possible cause of the decline in the red squirrel in the UK? Journal of Applied Ecology, 37 (2000), 997–1012. [27] E. Sheehy and C. Lawton, Population crash in an invasive species following the recovery of a native predator: The case of the American grey squirrel and the European pine marten in Ireland, Biodiversity and Conservatio, 23 (2014), 753-774.  doi: 10.1007/s10531-014-0632-7. [28] E. Sheehy and C. Lawton, Predators of red and grey squirrels in their natural and introduced range, In Red Squirrels Ecology, Conservation Management in Europe, ESI, European Squirrel Initiative, 83–96. [29] E. Sheehy, D. B. O'Meara, C. O'Reilly, A. Smart and C. Lawton, C, A non-invasive approach to determining pine marten abundance and predation, European Journal of Wildlife Research, 60 (2013), 223–236. doi: 10.1007/s10344-013-0771-2. [30] E. Sheehy, C. Sutherland, C. O'Reilly and X. Lambin, The enemy of my enemy is my friend: Native pine marten recovery reverses the decline of the red squirrel by suppressing grey squirrel populations, Proceedings of the Royal Society B: Biological Sciences, 285 (2018). doi: 10.1098/rspb.2017.2603. [31] M. Shorten, M. Vizoso and C. Elton, Some aspects of the biology of the grey squirrel (Sciurus carolinensis) in Great Britain, Proceedings of the Zoological Society of London, 121 (1951), 427-459. [32] M. A. Steele and L. A. Wauters, Diet and food hoarding in Eastern grey squirrels (Sciurus carolinensis): Implications for an invasive advantage, In The Grey Squirrel Ecology and Managamente of an Invasive Species in Europe, ESI, European Squirrel Initiative, 97–114. [33] K. Thomas, D. M. Tompkins, A. W. Sainsbury, A. R. Wood, R. Dalziel, P. F. Nettleton and C. J. McInnes, A novel poxvirus lethal to red squirrels (Sciurus vulgaris), Journal of General Virology, 84 (2003), 3337-3341. [34] L. A. Wauters and J. Gurnell, The mechanism of replacement of red squirrels by grey squirrels: A test of the interference competition hypothesis, Ethology, 29 (1999), 1053-1071. [35] L. A. Wauters, J. Gurnell, A. Martinoli and G. Tosi, Interspecific competition between native eurasian red squirrels and alien grey squirrels: Does resource partitioning occur?, Behavioral Ecology and Sociobiology, 52 (2002), 332-341.  doi: 10.1007/s00265-002-0516-9. [36] L. A. Wauters, P. W. W. Lurz and J. Gurnell, Interspecific effects of grey squirrels (Sciurus carolinensis) on the space use and population demography of red squirrels (Sciurus vulgaris) in conifer plantations, Ecological Research, 15 (2000), 271-284. [37] L. A. Wauters, G. Tosi and J. Gurnell, Interspecific competition in tree squirrels: do introduced grey squirrels (Sciurus carolinensis) deplete tree seeds hoarded by red squirrels (S. vulgaris)?, Behavioral Ecology and Sociobiology, 51 (2002), 360-367.  doi: 10.1007/s00265-001-0446-y. [38] A. Zalewski and W. Jedrzejewski, Spatial organisation and dynamics of the pine marten (Martes martes) population in Białowieza Forest (E Poland) compared with other European woodlands, Ecography, 29 (2006), 31-43.

show all references

##### References:
 [1] F. Barbara, V. La Morgia, V. Parodi, G. Toscano and E. Venturino Analysis of the incidence of poxvirus on the dynamics between red and grey squirrels, Mathematics, 6 (2018), 113. doi: 10.3390/math6070113. [2] S. Bertolino, Attivazione di un Progetto di Monitoraggio Estansivo ed Intensivo Dello Scoiattolo Grigio (Sciurus Carolinensis) in Piemonte, Technical report, Regione Piemonte, 2004. [3] S. Bertolino and P. Genovesi, The application of the European strategy on invasive alien species: An example with introduced squirrels, Hystrix, the Italian Journal of Mammalogy, 16 (2005), 59-69. [4] S. Bertolino, A. Martinoli and L. A Wauters, Risk Assessment for Sciurus carolinensis (grey squirrel), In H. Roy, editor, Invasive Alien Species-Framework for the Identification of Invasive Species of EU Concern, Natural Environment Research Council, (2014), 298. [5] L. Boitani, S. Lovari and A. Vigna Taglianti, Fauna d'Italia. Mammalia III: Carnivora-Artiodactyla, Edizioni Calderini de Il Sole-24 Ore, Bologna, 2003. [6] F. Cassola, Sciurus carolinensis, eastern gray squirrel, The IUCN Red List of Threatened Species, (2018), 1–8. [7] J. Chantrey, T. D. Dale, J. M. Read, S. White, F. Whitfield, D. Jones, C. J. McInnes and M. Begon, European red squirrel population dynamics driven by squirrelpox at a gray squirrel invasion interface, Ecology and Evolution, 4 (2014), 3788-3799.  doi: 10.1002/ece3.1216. [8] John Gurnell, The effects of food availability and winter weather on the dynamics of a grey squirrel population in Southern England, Journal of Applied Ecology, 33 (1996), 325-338.  doi: 10.2307/2404754. [9] J.Gurnell, The Natural History of Squirrels, Helm, London. [10] J. Gurnell, Squirrel numbers and the abundance of tree seeds, Mammal Review, 13 (1983), 133-148.  doi: 10.1111/j.1365-2907.1983.tb00274.x. [11] J. Gurnell, L. A. Wauter, P. W. W. Lurz and G. Tosi, Alien species and interspecific competition: Effects of introduced eastern grey squirrels on red squirrel population dynamics, Journal of Animal Ecology, 73 (2004), 26-35.  doi: 10.1111/j.1365-2656.2004.00791.x. [12] S. Harris, P. Morris, S. Wray and D. Yalden, A Review of British Mammals: Population Estimates and Conservation Status of British Mammals Other than Cetaceans, Technical report, Joint Nature Conservation Committee, Peterborough, 1995. [13] S. Harris, C. D. Soulsbury and G. Iossa, Is Culling Grey Squirrels a Viable Tactic to Conserve Red Squirrel Populations?, Technical report, Advocates for animals, 2006. [14] R. E. Kenward and J. M. Tonkin, Red and grey squirrels: Some behavioural and biometric differences, Journal of Zoology, 209 (1986), 279-281.  doi: 10.1111/j.1469-7998.1986.tb03583.x. [15] P. W. W. Lurz, S. P. Rushton, L. A. Wauters, S. Bertolino, I. Currado, P. Mazzoglio and M. D. F. Shirley, Predicting grey squirrel expansion in North Italy: A spatially explicit modelling approach, Landscape Ecology, 16 (2001), 407-420. [16] J. L. MacPherson, H. Denman, D. Tosh, C. McNicol and E. C. Halliwell, A review of the current evidence for impacts of the pine marten (Martes martes) on non-native and native squirrel populations, In The Grey Squirrel Ecology and Managamente of an Invasive Species in Europe, ESI, European Squirrel Initiative, 289–304. [17] A. Martinoli, S. Bertolino, D. G. Preatoni, A. Balduzzi, A. Marsan, P. Genovesi, G. Tosi and L. A. Wauters, Headcount 2010: The multiplication of the grey squirrel populations introduced to Italy, Hystrix, the Italian Journal of Mammalogy, 21 (2010), 127-136. [18] C. J. McInnes, A. R. Wood, K. Thomas, A. W. Sainsbury, J. Gurnell, F. J. Dein and P. F. Nettleton, Genomic characterization of a novel poxvirus contributing to the decline of the red squirrel (Sciurus vulgaris) in the UK, Journal of General Virology, 87, 2115–2125. [19] A. D. Middleton, 38. The Ecology of the American Grey Squirrel (Sciurus carolinensis Gmelin) in. the British Isles, Proceedings of the Zoological Society of London, 100 (1930), 809-843. [20] S. D. Montgomery, J. B. Whelan and H. S. Mosby, Bioenergetics of a woodlot gray squirrel population, The Journal of Wildlife Management, 39 (1975), 709-717.  doi: 10.2307/3800232. [21] H. S. Mosby, The influence of hunting on the population dynamics of a woodlot gray squirrel population, The Journal of Wildlife Management, 33 (1969), 59-73.  doi: 10.2307/3799650. [22] A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in Britain, Proceedings of the Royal Society of London. Series B, Biological sciences, 238 (1989), 113-125.  doi: 10.1098/rspb.1989.0070. [23] J. C. Reynolds, Details of the geographic replacement of the red squirrel (Sciurus vulgaris) by the grey squirrel (Sciurus carolinensis) in eastern england, The Journal of Animal Ecology, 54 (1985), 149–162. [24] C. Romeo, L. A. Wauters and N. Ferrari, 11. Parasite of grey squirrels: An additional threat to red squirrels in Italy?, In The Grey Squirrel Ecology and Managamente of an Invasive Species in Europe, ESI, European Squirrel Initiative, (2016), 193–209. [25] J. J. Rowe, Squirrel management, Mammal Review, 13 (1983), 173-181.  doi: 10.1111/j.1365-2907.1983.tb00279.x. [26] S. P. Rushton, P. W. W. Lurz, J. Gurnell, R. Fuller, Modelling the spatial dynamics of parapoxvirus disease in red and grey squirrels: A possible cause of the decline in the red squirrel in the UK? Journal of Applied Ecology, 37 (2000), 997–1012. [27] E. Sheehy and C. Lawton, Population crash in an invasive species following the recovery of a native predator: The case of the American grey squirrel and the European pine marten in Ireland, Biodiversity and Conservatio, 23 (2014), 753-774.  doi: 10.1007/s10531-014-0632-7. [28] E. Sheehy and C. Lawton, Predators of red and grey squirrels in their natural and introduced range, In Red Squirrels Ecology, Conservation Management in Europe, ESI, European Squirrel Initiative, 83–96. [29] E. Sheehy, D. B. O'Meara, C. O'Reilly, A. Smart and C. Lawton, C, A non-invasive approach to determining pine marten abundance and predation, European Journal of Wildlife Research, 60 (2013), 223–236. doi: 10.1007/s10344-013-0771-2. [30] E. Sheehy, C. Sutherland, C. O'Reilly and X. Lambin, The enemy of my enemy is my friend: Native pine marten recovery reverses the decline of the red squirrel by suppressing grey squirrel populations, Proceedings of the Royal Society B: Biological Sciences, 285 (2018). doi: 10.1098/rspb.2017.2603. [31] M. Shorten, M. Vizoso and C. Elton, Some aspects of the biology of the grey squirrel (Sciurus carolinensis) in Great Britain, Proceedings of the Zoological Society of London, 121 (1951), 427-459. [32] M. A. Steele and L. A. Wauters, Diet and food hoarding in Eastern grey squirrels (Sciurus carolinensis): Implications for an invasive advantage, In The Grey Squirrel Ecology and Managamente of an Invasive Species in Europe, ESI, European Squirrel Initiative, 97–114. [33] K. Thomas, D. M. Tompkins, A. W. Sainsbury, A. R. Wood, R. Dalziel, P. F. Nettleton and C. J. McInnes, A novel poxvirus lethal to red squirrels (Sciurus vulgaris), Journal of General Virology, 84 (2003), 3337-3341. [34] L. A. Wauters and J. Gurnell, The mechanism of replacement of red squirrels by grey squirrels: A test of the interference competition hypothesis, Ethology, 29 (1999), 1053-1071. [35] L. A. Wauters, J. Gurnell, A. Martinoli and G. Tosi, Interspecific competition between native eurasian red squirrels and alien grey squirrels: Does resource partitioning occur?, Behavioral Ecology and Sociobiology, 52 (2002), 332-341.  doi: 10.1007/s00265-002-0516-9. [36] L. A. Wauters, P. W. W. Lurz and J. Gurnell, Interspecific effects of grey squirrels (Sciurus carolinensis) on the space use and population demography of red squirrels (Sciurus vulgaris) in conifer plantations, Ecological Research, 15 (2000), 271-284. [37] L. A. Wauters, G. Tosi and J. Gurnell, Interspecific competition in tree squirrels: do introduced grey squirrels (Sciurus carolinensis) deplete tree seeds hoarded by red squirrels (S. vulgaris)?, Behavioral Ecology and Sociobiology, 51 (2002), 360-367.  doi: 10.1007/s00265-001-0446-y. [38] A. Zalewski and W. Jedrzejewski, Spatial organisation and dynamics of the pine marten (Martes martes) population in Białowieza Forest (E Poland) compared with other European woodlands, Ecography, 29 (2006), 31-43.
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
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