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January  2021, 26(1): 217-267. doi: 10.3934/dcdsb.2020328

Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence

Alex P. Farrell 1,, and  Horst R. Thieme 2,

1. 

Department of Mathematics, The University of Arizona, Tucson, AZ 85721-0089, USA
2. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA

* Corresponding author

Received  March 2020 Revised  October 2020 Published  January 2021 Early access  November 2020

To underpin the concern that environmental change can flip an ecosystem from stable persistence to sudden total collapse, we consider a class of so-called ecoepidemic models, predator – prey/host – parasite systems, in which a base species is prey to a predator species and host to a micro-parasite species. Our model uses generalized frequency-dependent incidence for the disease transmission and mass action kinetics for predation.

We show that a large variety of dynamics can arise, ranging from dynamic persistence of all three species to either total ecosystem collapse caused by high transmissibility of the parasite on the one hand or to parasite extinction and prey-predator survival due to low parasite transmissibility on the other hand. We identify a threshold parameter (tipping number) for the transition of the ecosystem from uniform prey/host persistence to total extinction under suitable initial conditions.

Keywords: Coexistence, multiple interior equilibria, bistability, homogeneous incidence, infection ratio, environmental change, climate change, ecology, epidemiology.
Mathematics Subject Classification: Primary: 92D30, 92D40, 34C11.
Citation: Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328
References:
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S. P. Bera, A. Maiti and G. P. Samanta, Prey-predator model with infection in both prey and predator, Filomat, 29 (2015), 1753-1767.  doi: 10.2298/FIL1508753B.

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E. Cagliero and E. Venturino, Ecoepidemics with infected prey in herd defence: The harmless and toxic cases, Int. J. Comp. Math., 93 (2016), 108-127.  doi: 10.1080/00207160.2014.988614.

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J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlin. Anal., 36 (1999), 747-766.  doi: 10.1016/S0362-546X(98)00126-6.

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J. Chattopadhyay and S. Pal, Viral infection on phytoplankton zooplankton system a mathematical model, Ecological Modeling, 151 (2002), 15-28.  doi: 10.1016/S0304-3800(01)00415-X.

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show all references

References:
[1]

F. Abbona and E. Venturino, An eco-epidemic model for infectious keratoconjunctivitis caused by mycoplasma conjunctivae in domestic and wild herbivores, with possible vaccination strategies, Math. Methods Appl. Sci., 41 (2018), 2269-2280.  doi: 10.1002/mma.4209.

[2]

O. Arino, A. El abdllaoui, J. Mikram and J. Chattopadhyay, Infection in prey population may act as a biological control in ratio-dependent predator-prey models, Nonlinearity, 17 (2004), 1101-1116.  doi: 10.1088/0951-7715/17/3/018.

[3]

B. S. Attili and S. F. Mallak, Existence of limit cycles in a predator-prey system with a functional response of the form arctan(ax), Commun. Math. Anal., 1 (2006), 33-40. 

[4]

N. Bairagi, P. K. Roy and J. Chattopadhyay, Role of infection on the stability of a predator-prey system with several response functionsa comparative study, J. Theo. Biol., 248 (2007), 10-25.  doi: 10.1016/j.jtbi.2007.05.005.

[5]

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.

[6]

E. Beltrami and T. O. Carroll, Modeling the role of viral disease in recurrent phytoplankton bloom, Journal of Mathematical Biology, 32 (1994), 857-863.  doi: 10.1007/BF00168802.

[7]

S. P. Bera, A. Maiti and G. P. Samanta, Prey-predator model with infection in both prey and predator, Filomat, 29 (2015), 1753-1767.  doi: 10.2298/FIL1508753B.

[8]

E. Cagliero and E. Venturino, Ecoepidemics with infected prey in herd defence: The harmless and toxic cases, Int. J. Comp. Math., 93 (2016), 108-127.  doi: 10.1080/00207160.2014.988614.

[9]

J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlin. Anal., 36 (1999), 747-766.  doi: 10.1016/S0362-546X(98)00126-6.

[10]

J. Chattopadhyay and S. Pal, Viral infection on phytoplankton zooplankton system a mathematical model, Ecological Modeling, 151 (2002), 15-28.  doi: 10.1016/S0304-3800(01)00415-X.

[11]

J. Chattopadhyay, S. Pal and A. El Abdllaoui, Classical predator-prey system with infection of prey populationa mathematical model, Math. Meth. in the App. Sci., 26 (2003), 1211-1222.  doi: 10.1002/mma.414.

[12]

Y. Chen and Y. Wen, Impact on the predator population while lethal disease spreads in the prey, Math. Meth. in App. Sci., 39 (2016), 2883-2895.  doi: 10.1002/mma.3737.

[13]

J. P. Collins, Amphibian decline and extinction: What we know and what we need to learn, Dis. Aquat. Org., 92 (2010), 93-99.  doi: 10.3354/dao02307.

[14] J. P. Collins, M. L. Crump and T. E. Lovejoy III, Extinction in our Times: Global Amphibian Decline, Oxford University Press, 2009. 
[15]

K. P. Das and J. Chattopadhyay, A mathematical study of a predator-prey model with disease circulating in the both populations, Int. J. Biomath., 8 (2015), 1550015, 27 pp. doi: 10.1142/S1793524515500151.

[16]

L. M. E. de Assis, E. Massad, R. A. de Assis, S. R. Martorano and E. Venturino, A mathematical model for bovine tuberculosis among buffaloes and lions in the kruger national park, Mathematical Methods in the Applied Sciences, 41 (2018), 525-543.  doi: 10.1002/mma.4568.

[17]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[18]

T. Dhirasakdanon and H. Thieme, Stability of the endemic coexistence equilibrium for one host and two parasites, Mathematical Modelling of Natural Phenomena, 5 (2010), 109-138.  doi: 10.1051/mmnp/20105606.

[19]

D. E. Docherty, C. U. Meteyer, J. Wang, J. Mao, S. T. Case and V. G. Chinchar, Diagnostic and molecular evaluation of three iridovirus-associated salamander mortality events, J. Wildl. Dis., 39 (2003), 556-566.  doi: 10.7589/0090-3558-39.3.556.

[20]

A. Farrell, Prey-Predator-Parasite: An Ecosystem Model with Fragile Persistence, Thesis (Ph.D.)-Arizona State University. 2017,238 pp.

[21]

A. P. Farrell, J. P. Collins, A. L. Greer and H. R. Thieme, Do fatal infectious diseases eradicate host species?, J. Math. Biol., 77 (2018), 2103-2164.  doi: 10.1007/s00285-018-1249-3.

[22]

A. P. Farrell, J. P. Collins, A. L. Greer and H. R. Thieme, Times from infection to disease-induced death and their influence on final population sizes after epidemic outbreaks, Bull. Math. Biol., 80 (2018), 1937-1961.  doi: 10.1007/s11538-018-0446-y.

[23]

A. Friedman and A.-A. Yakubu, Host demographic Allee effect, fatal disease, and migration: Persistence or extinction, SIAM J. Appl. Math., 72 (2012), 1644-1666.  doi: 10.1137/120861382.

[24]

J. Gani and R. J. Swift, Prey-predator models with infected prey and predators, Disc. and Cont. Dyn. Sys., 33 (2013), 5059-5066.  doi: 10.3934/dcds.2013.33.5059.

[25]

W. M. Getz and J. Pickering, Epidemic models: Thresholds and population regulation, Am. Nat., 121 (1983), 892-898.  doi: 10.1086/284112.

[26]

M. Ghosh and X.-Z. Li, Mathematical modelling of prey-predator interaction with disease in prey, Int. J. Comp. Sci. Math., 7 (2016), 443-458.  doi: 10.1504/IJCSM.2016.080075.

[27]

G. Gimmelli, B. W. Kooi and E. Venturino, Ecoepidemic models with prey group defense and feeding saturation, Ecol. Complex., 22 (2015), 50-58.  doi: 10.1016/j.ecocom.2015.02.004.

[28]

M. J. Gray and V. G. Chinchar, Ranaviruses: Lethal Pathogens of Ectothermic Vertebrates, Springer, 2015.

[29]

M. J. Gray, D. L. Miller and J. T. Hoverman, Ecology and pathology of amphibian ranaviruses, Dis. Aquat. Organ., 87 (2009), 243-266.  doi: 10.3354/dao02138.

[30]

D. E. Green, K. A. Converse and A. K. Schrader, Epizootiology of sixty-four amphibian morbidity and mortality events in the USA, 1996-2001, Ann. N. Y. Acad. Sci., 969 (2002), 323-339.  doi: 10.1111/j.1749-6632.2002.tb04400.x.

[31]

D. Greenhalgh and R. Das, An SIRS epidemic model with a contact rate depending on population density, Math. Pop. Dyn.: Anal. of Hetero., Vol. One: Theory of Epidemics, 92 (1995), 79-101. 

[32]

A. L. Greer, C. J. Briggs and J. P. Collins, Testing a key assumption of host-pathogen theory: Density and disease transmission, Oikos, 117 (2008), 1667-1673.  doi: 10.1111/j.1600-0706.2008.16783.x.

[33]

K. P. Hadeler, K. Dietz and M. Safan, Case fatality models for epidemics in growing populations, Math. Biosci., 281 (2016), 120-127.  doi: 10.1016/j.mbs.2016.09.007.

[34]

K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol., 27 (1989), 609-631.  doi: 10.1007/BF00276947.

[35]

J. K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Company, Inc., 1980.

[36]

L. Han, Z. Ma and H. W. Hethcote, Four predator prey models with infectious diseases, Math. Comp. Modeling, 34 (2001), 849-858.  doi: 10.1016/S0895-7177(01)00104-2.

[37]

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Table 1.  Summary of the dynamics of the host-parasite subsystem when $ \xi $ is decreasing
$ ^{{\dagger}} $ means that this event only occurs if the corresponding parameter inequality is strict or $ \xi $ is strictly decreasing; GAS stands for "globally asymptotically stable."
Parameter Values Dynamics
$ {{\sigma}}\le \frac{{{\mu}}}{h'(0)} $ $ S(0)>0 \Longrightarrow r(t)\to 0, S(t)\to K $
$ \frac{{{\mu}}}{h'(0)}<{{\sigma}}\le \frac{{{\mu}}+g(0)}{h'(0)} $ no $ (0,r^\circ) $, $ (S^*,r^*) $ GAS$ ^{{\dagger}} $ for $ (0,\infty)^2 $
$ \frac{{{\mu}}+g(0)}{h'(0)}<{{\sigma}}<\frac{g(0)}{h(g(0)/{{\mu}})} $ $ \exists (0,r^\circ) $, $ (S^*,r^*) $ GAS for $ (0,\infty)^2 $
$ \frac{g(0)}{h(g(0)/{{\mu}})} \le {{\sigma}}<\frac{{{\mu}}+g(0)}{h(\infty)} $ $ \exists (0,r^\circ) $, $ r(0)>0 \Longrightarrow S(t)\to 0 $
$ \frac{{{\mu}}+g(0)}{h(\infty)}\le {{\sigma}} $ $ r(0)> 0 \Longrightarrow S(t)\to 0, (r(t)\to\infty)^{{\dagger}} $
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$ ^{{\dagger}} $ means that this event only occurs if the corresponding parameter inequality is strict or $ \xi $ is strictly decreasing; GAS stands for "globally asymptotically stable."
Parameter Values Dynamics
$ {{\sigma}}\le \frac{{{\mu}}}{h'(0)} $ $ S(0)>0 \Longrightarrow r(t)\to 0, S(t)\to K $
$ \frac{{{\mu}}}{h'(0)}<{{\sigma}}\le \frac{{{\mu}}+g(0)}{h'(0)} $ no $ (0,r^\circ) $, $ (S^*,r^*) $ GAS$ ^{{\dagger}} $ for $ (0,\infty)^2 $
$ \frac{{{\mu}}+g(0)}{h'(0)}<{{\sigma}}<\frac{g(0)}{h(g(0)/{{\mu}})} $ $ \exists (0,r^\circ) $, $ (S^*,r^*) $ GAS for $ (0,\infty)^2 $
$ \frac{g(0)}{h(g(0)/{{\mu}})} \le {{\sigma}}<\frac{{{\mu}}+g(0)}{h(\infty)} $ $ \exists (0,r^\circ) $, $ r(0)>0 \Longrightarrow S(t)\to 0 $
$ \frac{{{\mu}}+g(0)}{h(\infty)}\le {{\sigma}} $ $ r(0)> 0 \Longrightarrow S(t)\to 0, (r(t)\to\infty)^{{\dagger}} $
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