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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 & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328
References:
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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.  Google Scholar

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[6]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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[11]

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[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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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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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