doi: 10.3934/dcdsb.2021158

A viral transmission model for foxes-cottontails-hares interaction: Infection through predation

1. 

Dipartimento di Matematica "Giuseppe Peano", Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy

2. 

Department of Life Sciences and Systems Biology, University of Turin, via Accademia Albertina 13, 10123 Torino, Italy

* Corresponding author: Ezio Venturino

Received  November 2020 Revised  April 2021 Published  June 2021

Fund Project: The last author has been partially supported by the project "Metodi numerici nelle scienze applicate" of the Dipartimento di Matematica "Giuseppe Peano" of the Università di Torino
EV is a member of the research group GNCS

The Eastern cottontail Sylvilagus floridanus is a lagomorph native to North America, introduced in Italy since the 1960s. In Central and Northern Italy, the cottontail overlaps its range with the native European hare Lepus europaeus and affects the predator-prey dynamics of native hares and foxes. Field data indicate that the cottontail is susceptible to infection by the European brown hare syndrome (EBHS) virus. Although the real role of cottontails and native foxes in the spreading of EBHS viruses is yet uncertain, we present a cottontail-hare-fox model including possible effects of EBHS, imported by foxes, through environmental contamination. A rather complete map of the possible system equilibria and their mutual relationship and transition is established.

Citation: Simona Viale, Elisa Caudera, Sandro Bertolino, Ezio Venturino. A viral transmission model for foxes-cottontails-hares interaction: Infection through predation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021158
References:
[1]

G. Amori, L. Contoli and A. Nappi, Fauna d'Italia, Mammalia II: Erinaceomorpha, Soricomorpha, Lagomorpha, Rodentia, Edizioni Calderini, Bologna, Italia, 2008. Google Scholar

[2]

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. Google Scholar

[3]

S. BertolinoN. Cordero di Montezemolo and A. Perrone, Daytime habitat selection by introduced eastern cottontail, (Sylvilagus Floridanus) and Native European hare (Lepus Europaeus) in Northern Italy, Zool. Sci., 28 (2011b), 414-419.   Google Scholar

[4]

S. BertolinoN. Cordero di Montezemolo and A. Perrone, Habitat use of coexisting introduced eastern cottontail and native European hare., Mamm. Biol., 78 (2013), 235-240.   Google Scholar

[5]

S. BertolinoL. HofmannováM. Girardello and D. Modry, Richness, origin and structure of an Eimeria community in a population of Eastern cottontail (Sylvilagus Floridanus) introduced into Italy, Parasitology, 137 (2010), 1179-1186.   Google Scholar

[6]

S. BertolinoA. PerroneL. Gola and R. Viterbi, Population density and habitat use of introduced eastern cottontail (Sylvilagus Floridanus) in Comparison with the Native European hare (Lepus Europaeus), Zool. Stud., 50 (2011a), 315-326.   Google Scholar

[7]

L. Boitani, S. Lovari and A. Vigna Taglianti, Fauna d'Italia. Mammalia III, Carnivora-Artiodactyla, Edizioni Calderini, Bologna, Italia, 2003. Google Scholar

[8]

E. Caudera, S. Viale, S. Bertolino, J. Cerri and E. Venturino, A mathematical model supporting a hyperpredation effect in the apparent competition between invasive Eastern cottontail and native European hare, Bulletin of Mathematical Biology, 83 (2021), 51. doi: 10.1007/s11538-021-00873-9.  Google Scholar

[9]

E. CauderaS. VialeS. Bertolino and E. Venturino, A European Brown Hare Syndrome model within a foxes-cottontails-hares interaction system, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 66 (2020), 347-384.   Google Scholar

[10]

J. CerriM. Ferretti and S. Bertolino, Rabbits killing hares: An invasive mammal modifies native predator-prey dynamics, Animal Conservation, 20 (2017), 511-519.  doi: 10.1111/acv.12343.  Google Scholar

[11]

M. ChiariS. MolinariP. CavadiniB. BertasiM. ZanoniL. Capucci and A. Lavazza, Red foxes Vulpes vulpes feeding brown hares Lepus europaeus infected by European brown hare syndrome virus (EBHSv) might be involved in the spread of the virus, European Journal of Wildlife Research, 62 (2016), 761-765.   Google Scholar

[12]

E. Chinchio, M. Crotta, C. Romeo, J. A. Drewe, J. Guitian and N. Ferrari, Invasive alien species and disease risk: An open challenge in public and animal health., PLoS Pathog, 16 (2020), e1008922. doi: 10.1371/journal.ppat.1008922.  Google Scholar

[13]

A. D'AngeloJ. CerriP. CavadiniA. LavazzaL. Capucci and M. Ferretti, The Eastern cottontail Sylvilagus floridanus in Tuscany (Central Italy): Weak evidence for its role as a host of EBHSV and RHDV., Hystrix, the Italian Journal of Mammalogy, 30 (2019), 8-11.   Google Scholar

[14]

P. DaszakA. A. Cunningham and A. D. Hyatt, Emerging infectious diseases of wildlife–threats to biodiversity and human health, Science, 287 (2000), 443-449.   Google Scholar

[15]

P. DoriM. Scalisi and E. Mori, "An American near Rome"... and not only! Presence of the Eastern Cottontail in Central Italy and Potential Impacts on the Endemic and Vulnerable Apennine Hare, Mammalia, 83 (2019), 307-312.   Google Scholar

[16]

A. M. Dunn and M. J. Hatcher, Parasites and biological invasions: Parallels, interactions, and control, Trends Parasitol, 31 (2015), 189-199.   Google Scholar

[17]

K. Frölich and A. Lavazza, European brown hare syndrome, Lagomorph Biology, Springer, Berlin, Heidelberg, (2008), 253–261. Google Scholar

[18]

J. Gurnell, L. A. Wauters, 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), 2635. Google Scholar

[19]

A. Lavazza, P. Cavadini, I. Barbieri, P. Tizzani, A. Pinheiro, J. Abrantes, P. J. Esteves, G. Grilli, E. Gioia, M. Zanoni, P. G. Meneguz, J. Guitton, S. Marchandeau, M. Chiari and L. Cappucci, Field and experimental data indicate that the eastern cottontail Sylvilagus floridanus is susceptible to infection with European brown hare syndrome (EBHS) virus and not with rabbit haemorrhagic disease (RHD) virus, Veterinary Research, 46 (2015), 13. Google Scholar

[20]

A. Lavazza, V. Guberti, M. Ferri, M. L. Zanni, G. Poglayen, A. Nardin and L. Capucci, Epidemiology of European brown hare syndrome (EBHS) in Modena province (North Italy), In Proceedings 4th International Congress of Veterinary Virology, ESVV, 24 (1997), 27. Google Scholar

[21]

V. La MorgiaE. Travaglia and E. 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, 25 (2020), 2023-2040.  doi: 10.3934/dcdsb.2019200.  Google Scholar

[22]

A. LoyG. Aloise and L. Ancillotto, Mammals of Italy: an annotated checklist, Hystrix Ital. J. Mamm., 30 (2019), 87-106.   Google Scholar

[23]

G. NugentB. M. Buddle and G. Knowles, Epidemiology and control of Mycobacterium bovis infection in brushtail possums Trichosurus vulpecula, the primary wildlife host of bovine tuberculosis, New Zealand. N Z Vet J., 63 (2015), 28-41.   Google Scholar

[24]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[25]

C. RomeoC. J. McInnesT. D. DaleC. ShuttleworthS. BertolinoL. A. Wauters and N. Ferrari, Disease, invasions and conservation: No evidence of squirrelpox virus in grey squirrels introduced to Italy, Animal Conservation, 22 (2018), 14-23.   Google Scholar

[26]

M. SieberH. Malchow and F. Hilker, Disease-induced modification of prey competition in eco-epidemiological models, Ecol. Complex., 18 (2014), 74-82.   Google Scholar

[27]

D. SimberloffJ. L. MartinP. GenovesiV. MarisD. A. WardleJ. AronsonF. CourchampB. GalilE. García-BerthouM. PascalP. PyšekR. SousaE. Tabacchi and M. Vilà, Impacts of biological invasions: what's what and the way forward., Trends Ecol. Evol., 28 (2013), 58-66.   Google Scholar

[28]

A. StraussA. White and M. Boots, Invading with biological weapons: The importance of disease-mediated invasions, Functional Ecology, 26 (2012), 1249-1261.   Google Scholar

[29]

P. TizzaniS. CatalanoL. RossiP. J. Duignan and P. G. Meneguz, Invasive species and their parasites: Eastern cottontail rabbit Sylvilagus floridanus and Trichostrongylus affinis (Graybill, 1924) from Northwestern Italy, Parasitol. Res., 113 (2014), 1301-1303.   Google Scholar

[30]

D. M. TompkinsA. W. SainsburyP. NettletonD. Buxton and J. Gurnell, Parapoxvirus causes a deleterious disease in red squirrels associated with UK population declines, Proceedings of the Royal Society of London, Series B, Biological Sciences, 269 (2002), 529-533.   Google Scholar

[31]

D. M. TompkinsA. R. White and M. Boots, Ecological replacement of native red squirrels by invasive greys driven by disease, Ecology Letters, 6 (2003), 189-196.   Google Scholar

[32]

E. Venturino, The influence of diseases on Lotka-Volterra systems, Rocky Mountain J. Math., 24 (1994), 381-402.  doi: 10.1216/rmjm/1181072471.  Google Scholar

[33]

J. R. WalshS. R. Carpenter and M. J. Vander Zanden, Invasive species triggers a massive loss of ecosystem services through a trophic cascade, Proceedings of the National Academy of Sciences of the United States of America, 113 (2016), 4081-4085.   Google Scholar

[34]

L. A. WautersJ. Gurnell and A. Martinoli, Interspecific competition between native Eurasian red squirrels and alien grey squirrels: does resource partitioning occur?, Behavioral Ecology and Sociobiology, 52 (2002), 332-341.   Google Scholar

show all references

References:
[1]

G. Amori, L. Contoli and A. Nappi, Fauna d'Italia, Mammalia II: Erinaceomorpha, Soricomorpha, Lagomorpha, Rodentia, Edizioni Calderini, Bologna, Italia, 2008. Google Scholar

[2]

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. Google Scholar

[3]

S. BertolinoN. Cordero di Montezemolo and A. Perrone, Daytime habitat selection by introduced eastern cottontail, (Sylvilagus Floridanus) and Native European hare (Lepus Europaeus) in Northern Italy, Zool. Sci., 28 (2011b), 414-419.   Google Scholar

[4]

S. BertolinoN. Cordero di Montezemolo and A. Perrone, Habitat use of coexisting introduced eastern cottontail and native European hare., Mamm. Biol., 78 (2013), 235-240.   Google Scholar

[5]

S. BertolinoL. HofmannováM. Girardello and D. Modry, Richness, origin and structure of an Eimeria community in a population of Eastern cottontail (Sylvilagus Floridanus) introduced into Italy, Parasitology, 137 (2010), 1179-1186.   Google Scholar

[6]

S. BertolinoA. PerroneL. Gola and R. Viterbi, Population density and habitat use of introduced eastern cottontail (Sylvilagus Floridanus) in Comparison with the Native European hare (Lepus Europaeus), Zool. Stud., 50 (2011a), 315-326.   Google Scholar

[7]

L. Boitani, S. Lovari and A. Vigna Taglianti, Fauna d'Italia. Mammalia III, Carnivora-Artiodactyla, Edizioni Calderini, Bologna, Italia, 2003. Google Scholar

[8]

E. Caudera, S. Viale, S. Bertolino, J. Cerri and E. Venturino, A mathematical model supporting a hyperpredation effect in the apparent competition between invasive Eastern cottontail and native European hare, Bulletin of Mathematical Biology, 83 (2021), 51. doi: 10.1007/s11538-021-00873-9.  Google Scholar

[9]

E. CauderaS. VialeS. Bertolino and E. Venturino, A European Brown Hare Syndrome model within a foxes-cottontails-hares interaction system, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 66 (2020), 347-384.   Google Scholar

[10]

J. CerriM. Ferretti and S. Bertolino, Rabbits killing hares: An invasive mammal modifies native predator-prey dynamics, Animal Conservation, 20 (2017), 511-519.  doi: 10.1111/acv.12343.  Google Scholar

[11]

M. ChiariS. MolinariP. CavadiniB. BertasiM. ZanoniL. Capucci and A. Lavazza, Red foxes Vulpes vulpes feeding brown hares Lepus europaeus infected by European brown hare syndrome virus (EBHSv) might be involved in the spread of the virus, European Journal of Wildlife Research, 62 (2016), 761-765.   Google Scholar

[12]

E. Chinchio, M. Crotta, C. Romeo, J. A. Drewe, J. Guitian and N. Ferrari, Invasive alien species and disease risk: An open challenge in public and animal health., PLoS Pathog, 16 (2020), e1008922. doi: 10.1371/journal.ppat.1008922.  Google Scholar

[13]

A. D'AngeloJ. CerriP. CavadiniA. LavazzaL. Capucci and M. Ferretti, The Eastern cottontail Sylvilagus floridanus in Tuscany (Central Italy): Weak evidence for its role as a host of EBHSV and RHDV., Hystrix, the Italian Journal of Mammalogy, 30 (2019), 8-11.   Google Scholar

[14]

P. DaszakA. A. Cunningham and A. D. Hyatt, Emerging infectious diseases of wildlife–threats to biodiversity and human health, Science, 287 (2000), 443-449.   Google Scholar

[15]

P. DoriM. Scalisi and E. Mori, "An American near Rome"... and not only! Presence of the Eastern Cottontail in Central Italy and Potential Impacts on the Endemic and Vulnerable Apennine Hare, Mammalia, 83 (2019), 307-312.   Google Scholar

[16]

A. M. Dunn and M. J. Hatcher, Parasites and biological invasions: Parallels, interactions, and control, Trends Parasitol, 31 (2015), 189-199.   Google Scholar

[17]

K. Frölich and A. Lavazza, European brown hare syndrome, Lagomorph Biology, Springer, Berlin, Heidelberg, (2008), 253–261. Google Scholar

[18]

J. Gurnell, L. A. Wauters, 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), 2635. Google Scholar

[19]

A. Lavazza, P. Cavadini, I. Barbieri, P. Tizzani, A. Pinheiro, J. Abrantes, P. J. Esteves, G. Grilli, E. Gioia, M. Zanoni, P. G. Meneguz, J. Guitton, S. Marchandeau, M. Chiari and L. Cappucci, Field and experimental data indicate that the eastern cottontail Sylvilagus floridanus is susceptible to infection with European brown hare syndrome (EBHS) virus and not with rabbit haemorrhagic disease (RHD) virus, Veterinary Research, 46 (2015), 13. Google Scholar

[20]

A. Lavazza, V. Guberti, M. Ferri, M. L. Zanni, G. Poglayen, A. Nardin and L. Capucci, Epidemiology of European brown hare syndrome (EBHS) in Modena province (North Italy), In Proceedings 4th International Congress of Veterinary Virology, ESVV, 24 (1997), 27. Google Scholar

[21]

V. La MorgiaE. Travaglia and E. 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, 25 (2020), 2023-2040.  doi: 10.3934/dcdsb.2019200.  Google Scholar

[22]

A. LoyG. Aloise and L. Ancillotto, Mammals of Italy: an annotated checklist, Hystrix Ital. J. Mamm., 30 (2019), 87-106.   Google Scholar

[23]

G. NugentB. M. Buddle and G. Knowles, Epidemiology and control of Mycobacterium bovis infection in brushtail possums Trichosurus vulpecula, the primary wildlife host of bovine tuberculosis, New Zealand. N Z Vet J., 63 (2015), 28-41.   Google Scholar

[24]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[25]

C. RomeoC. J. McInnesT. D. DaleC. ShuttleworthS. BertolinoL. A. Wauters and N. Ferrari, Disease, invasions and conservation: No evidence of squirrelpox virus in grey squirrels introduced to Italy, Animal Conservation, 22 (2018), 14-23.   Google Scholar

[26]

M. SieberH. Malchow and F. Hilker, Disease-induced modification of prey competition in eco-epidemiological models, Ecol. Complex., 18 (2014), 74-82.   Google Scholar

[27]

D. SimberloffJ. L. MartinP. GenovesiV. MarisD. A. WardleJ. AronsonF. CourchampB. GalilE. García-BerthouM. PascalP. PyšekR. SousaE. Tabacchi and M. Vilà, Impacts of biological invasions: what's what and the way forward., Trends Ecol. Evol., 28 (2013), 58-66.   Google Scholar

[28]

A. StraussA. White and M. Boots, Invading with biological weapons: The importance of disease-mediated invasions, Functional Ecology, 26 (2012), 1249-1261.   Google Scholar

[29]

P. TizzaniS. CatalanoL. RossiP. J. Duignan and P. G. Meneguz, Invasive species and their parasites: Eastern cottontail rabbit Sylvilagus floridanus and Trichostrongylus affinis (Graybill, 1924) from Northwestern Italy, Parasitol. Res., 113 (2014), 1301-1303.   Google Scholar

[30]

D. M. TompkinsA. W. SainsburyP. NettletonD. Buxton and J. Gurnell, Parapoxvirus causes a deleterious disease in red squirrels associated with UK population declines, Proceedings of the Royal Society of London, Series B, Biological Sciences, 269 (2002), 529-533.   Google Scholar

[31]

D. M. TompkinsA. R. White and M. Boots, Ecological replacement of native red squirrels by invasive greys driven by disease, Ecology Letters, 6 (2003), 189-196.   Google Scholar

[32]

E. Venturino, The influence of diseases on Lotka-Volterra systems, Rocky Mountain J. Math., 24 (1994), 381-402.  doi: 10.1216/rmjm/1181072471.  Google Scholar

[33]

J. R. WalshS. R. Carpenter and M. J. Vander Zanden, Invasive species triggers a massive loss of ecosystem services through a trophic cascade, Proceedings of the National Academy of Sciences of the United States of America, 113 (2016), 4081-4085.   Google Scholar

[34]

L. A. WautersJ. Gurnell and A. Martinoli, Interspecific competition between native Eurasian red squirrels and alien grey squirrels: does resource partitioning occur?, Behavioral Ecology and Sociobiology, 52 (2002), 332-341.   Google Scholar

Figure 1.  Equilibrium $ E_4 $, attained for the parameter values $ r = \log(3) $, $ s = \log(4.5) $, $ u = \log(5) $, $ r_H = 0.3 $, $ r_U = 0.9 $, $ r_I = 0.4 $, $ m = \frac 27 $, $ \mu = 0.1 $, $ n = 4/5 $, $ p = 2/11 $, $ a = 0.2 $, $ b = 0.1 $, $ c = 0.2 $, $ h = 0.4 $, $ e = 0.91 $, $ c_{VV} = \log(3)-\frac 27 $, $ c_{VU} = c_{VV}-0.1 $, $ c_{UU} = \log(3)-\frac 27 $, $ c_{UV} = c_{UU}+0.1 $, $ c_{SS} = \log(4.5)/100-4/500 $, $ c_{LL} = \log(5)/30-2/330 $, $ \zeta = 0.4 $, $ \theta = 0.7 $, $ \xi = 3 $, $ \eta = 2 $ and initial conditions (29)
Figure 2.  Equilibrium $ E_7 $, attained for the parameter values $ r = \log(3) $, $ s = \log(4.5) $, $ u = \log(5) $, $ r_H = 0.3 $, $ r_U = 0.9 $, $ r_I = 0.4 $, $ m = 3 $, $ \mu = 0.1 $, $ n = 4/5 $, $ p = 2/11 $, $ a = 5 $, $ b = 3 $, $ c = 4 $, $ h = 4 $, $ e = 0.91 $, $ c_{VV} = \log(3)-\frac 27 $, $ c_{VU} = c_{VV}-0.1 $, $ c_{UU} = \log(3)-\frac 27 $, $ c_{UV} = c_{UU}+0.1 $, $ c_{SS} = \log(4.5)/100-4/500 $, $ c_{LL} = \log(5)/30-2/330 $, $ \zeta = 0.8 $, $ \theta = 0.7 $, $ \xi = 0.3 $, $ \eta = 0.2 $ and initial conditions (29). Interestingly, this equilibrium shows damped oscillations
Figure 3.  Equilibrium $ E_9 $. Coexistence is attained for the parameter values $ r = \log(3) $, $ s = \log(4.5) $, $ u = \log(5) $, $ r_H = 0.3 $, $ r_U = 0.9 $, $ r_I = 0.4 $, $ m = \frac 27 $, $ \mu = 0.1 $, $ n = 4/5 $, $ p = 2/11 $, $ a = 0.2 $, $ b = 0.1 $, $ c = 0.2 $, $ h = 0.4 $, $ e = 0.91 $, $ c_{VV} = \log(3)-\frac 27 $, $ c_{VU} = c_{VV}-0.1 $, $ c_{UU} = \log(3)-\frac 27 $, $ c_{UV} = c_{UU}+0.1 $, $ c_{SS} = \log(4.5)/100-4/500 $, $ c_{LL} = \log(5)/30-2/330 $, $ \zeta = 0.4 $, $ \theta = 0.7 $, $ \xi = 0.3 $, $ \eta = 0.2 $ and initial conditions (29)
Figure 7.  Picture of the whole bifurcation structure of the model (17) arising from the analysis of Section 4. In each node, the equilibrium name is reported, together with its nonvanishing populations. The red lines indicate the bifurcations that have been found only numerically. Bifurcation thresholds: $ r_{0, 1} = m $, $ \mu_{1, 2} = \zeta r_U-c_{UV}V_1-m $, $ s_{1, 4} = n+a V_1 $, $ u_{1, 6} = p+b V_1 $, $ s_{2, 4} = n+a V_2+c U_2 $, $ u_{2, 7} = p+bV_2+hU_2+\eta \frac{U_2}{V_2+U_2} $, $ \mu_{0, 2} = \zeta r_U-m $, $ s_{0, 3} = n $, $ u_{3, 8} = p+\xi $, $ u_{0, 5} = p $, $ r_{5, 6} = m-r_H be L_5 $, $ s_{5, 8} = n $, $ \mu_{5, 7} = \zeta r_U-m+\zeta r_I he L_5 $, $ \mu_{6, 7} = \zeta r_U-c_{UV} V_6-m+\zeta r_I he L_6 $, $ s_{6, 9} = n+a V_6 $, $ u_{4, 9} = p+bV_4+hU_4+\frac{\xi S_4+\eta U_4}{V_4+U_4+S_4} $, $ s_{7, 9} = n+aV_7+cU_7 $. The asymmetry in the graph could be interpreted biologically as follows. Consider for instance the absence of the arc $ E_8-E_4 $ while arc $ E_2-E_7 $ is present. Both apparently connect points with 2 and 3 populations, but in reality $ E_2 $ contains only foxes, susceptible $ V $ and infected $ U $, while $ E_8 $ has two different system populations, cottontails $ S $ and hares $ L $. A similar claim can be made for the pairs $ E_0-E_8 $ and $ E_0-E_2 $
Figure 4.  Transcritical bifurcation relating equilibria $ E_2 $ and $ E_0 $ of the system (17) in terms of the bifurcation parameter $ \mu \in [1, 4] $, the foxes virus-related mortality
Figure 5.  Transcritical bifurcation relating equilibria $ E_3 $ and $ E_4 $ of the system (17) in terms of the healthy foxes reproduction rate $ r\in [1, 3.5] $ as bifurcation parameter. The parameters are given by (28) with the exception of those in (31)
Figure 6.  Transcritical bifurcation connecting equilibria $ E_8 $ and $ E_9 $ of the system (17) for the bifurcation parameter given by the healthy foxes reproduction rate $ r \in [3, 6] $. The parameters are given by (28) with the exception of those in (32)
Table 1.  The model parameters and their meaning
Parameter Interpretation
$ r $ healthy foxes reproduction rate on other resources
$ r_U $ infected foxes reproduction rate on other resources
$ r_H $ healthy foxes reproduction rate on cottontails and hares
$ r_I $ infected foxes reproduction rate on cottontails and hares
$ m $ foxes natural mortality rate
$ c_{UU} $, $ c_{VV} $, $ c_{UV} $, $ c_{VU} $ foxes intraspecific competition coefficients
$ \zeta \le 1 $ foxes vertical virus transmission
$ \theta $ foxes transmission rate by feeding on infected cottontail
$ a $ healthy foxes predation rate on cottontails
$ b $ healthy foxes predation rate on hares
$ c $ infected foxes predation rate on cottontails
$ h $ infected foxes predation rate on hares
$ e $ foxes conversion coefficient of captured prey
$ \mu $ foxes mortality rate due to virus, possibly $ \mu=0 $
$ s $ cottontails reproduction rate
$ c_{SS} $ cottontails intraspecific competition
$ n $ cottontails natural mortality rate
$ u $ hares reproduction rate
$ c_{LL} $ hares intraspecific competition
$ p $ hares natural mortality rate
$ \xi $ hares infection, i.e. direct mortality rate,
by cottontails environment pollution
$ \eta $ hares infection, i.e. direct mortality rate,
by infected foxes environment pollution
Parameter Interpretation
$ r $ healthy foxes reproduction rate on other resources
$ r_U $ infected foxes reproduction rate on other resources
$ r_H $ healthy foxes reproduction rate on cottontails and hares
$ r_I $ infected foxes reproduction rate on cottontails and hares
$ m $ foxes natural mortality rate
$ c_{UU} $, $ c_{VV} $, $ c_{UV} $, $ c_{VU} $ foxes intraspecific competition coefficients
$ \zeta \le 1 $ foxes vertical virus transmission
$ \theta $ foxes transmission rate by feeding on infected cottontail
$ a $ healthy foxes predation rate on cottontails
$ b $ healthy foxes predation rate on hares
$ c $ infected foxes predation rate on cottontails
$ h $ infected foxes predation rate on hares
$ e $ foxes conversion coefficient of captured prey
$ \mu $ foxes mortality rate due to virus, possibly $ \mu=0 $
$ s $ cottontails reproduction rate
$ c_{SS} $ cottontails intraspecific competition
$ n $ cottontails natural mortality rate
$ u $ hares reproduction rate
$ c_{LL} $ hares intraspecific competition
$ p $ hares natural mortality rate
$ \xi $ hares infection, i.e. direct mortality rate,
by cottontails environment pollution
$ \eta $ hares infection, i.e. direct mortality rate,
by infected foxes environment pollution
Table 2.  Equilibria feasibility
Equilibrium Feasibility conditions
$ E_0 $ $ - $
$ E_1 $ $ r \ge m $
$ E_2^- $ (23), (24), (25)
$ E_2^{\pm} $ sufficent (26)
$ E_3 $ $ s \ge n $,
$ E_4 $ numerical
$ E_5 $ $ u \ge p $
$ E_6 $ (27)
$ E_7 $ numerical
$ E_8^+ $ sufficient: $ u\ge p+\xi $
$ E_8^{\pm} $ $ u<\xi $, $ (u-p)^2 \ge 4 c_{LL} S_8 (p+\xi-u) $
$ E_9 $ numerical
Equilibrium Feasibility conditions
$ E_0 $ $ - $
$ E_1 $ $ r \ge m $
$ E_2^- $ (23), (24), (25)
$ E_2^{\pm} $ sufficent (26)
$ E_3 $ $ s \ge n $,
$ E_4 $ numerical
$ E_5 $ $ u \ge p $
$ E_6 $ (27)
$ E_7 $ numerical
$ E_8^+ $ sufficient: $ u\ge p+\xi $
$ E_8^{\pm} $ $ u<\xi $, $ (u-p)^2 \ge 4 c_{LL} S_8 (p+\xi-u) $
$ E_9 $ numerical
Table 3.  Equilibria stability
Equilibria Stability conditions
$ E_0 $ $ m>r $, $ s>n $, $ u>p $
$ E_1 $ $ \max \left\{ { \frac {\zeta r_U - m - \mu} {c_{UV} } , \frac {s - n} a , \frac {u - p} b } \right\}< V_1 $
$ E_2 $ $ s< n+aV_2+cU_2, \quad u< p+bV_2+hU_2+\eta \frac {U_2}{U_2+V_2} $, (39)
$ E_3 $ $ n< s $, $ u< p + \xi $, (41)
$ E_4 $ $ u< p+bV_4+hU_4 + \frac{\xi S_4 + \eta U_4}{V_4+U_4+S_4} $, (43)
$ E_5 $ $ r+r_H ebL_5< m $, $ \zeta r_U + \zeta r_I he L_5< m+\mu $, $ s< n $
$ E_6 $ $ \zeta r_U-c_{UV}V_6+\zeta r_I he L_6< m+\mu $, $ s< n + a V_6 $
$ E_7 $ $ s< n + aV_7 + cU_7 $, (47)
$ E_8 $ $ \frac{\xi S_8}{(S_8+L_8)^2}< c_{LL} $, $ n< s $, (49)
$ E_9 $ numerical
Equilibria Stability conditions
$ E_0 $ $ m>r $, $ s>n $, $ u>p $
$ E_1 $ $ \max \left\{ { \frac {\zeta r_U - m - \mu} {c_{UV} } , \frac {s - n} a , \frac {u - p} b } \right\}< V_1 $
$ E_2 $ $ s< n+aV_2+cU_2, \quad u< p+bV_2+hU_2+\eta \frac {U_2}{U_2+V_2} $, (39)
$ E_3 $ $ n< s $, $ u< p + \xi $, (41)
$ E_4 $ $ u< p+bV_4+hU_4 + \frac{\xi S_4 + \eta U_4}{V_4+U_4+S_4} $, (43)
$ E_5 $ $ r+r_H ebL_5< m $, $ \zeta r_U + \zeta r_I he L_5< m+\mu $, $ s< n $
$ E_6 $ $ \zeta r_U-c_{UV}V_6+\zeta r_I he L_6< m+\mu $, $ s< n + a V_6 $
$ E_7 $ $ s< n + aV_7 + cU_7 $, (47)
$ E_8 $ $ \frac{\xi S_8}{(S_8+L_8)^2}< c_{LL} $, $ n< s $, (49)
$ E_9 $ numerical
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