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

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

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.

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:
[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

[2]

O. ArinoA. El abdllaouiJ. 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.  Google Scholar

[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.   Google Scholar

[4]

N. BairagiP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. P. BeraA. 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]

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.  Google Scholar

[11]

J. ChattopadhyayS. 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.  Google Scholar

[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

[14] J. P. CollinsM. L. Crump and T. E. Lovejoy III, Extinction in our Times: Global Amphibian Decline, Oxford University Press, 2009.   Google Scholar
[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 AssisE. MassadR. A. de AssisS. 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. DochertyC. U. MeteyerJ. WangJ. MaoS. 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. FarrellJ. P. CollinsA. 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. FarrellJ. P. CollinsA. 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. GimmelliB. 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. GrayD. 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. GreenK. 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]

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.   Google Scholar

[32]

A. L. GreerC. 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.  Google Scholar

[33]

K. P. HadelerK. 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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

L. HanZ. 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.  Google Scholar

[37]

L. Han and A. Pugliese, Epidemics in two competing species, Nonlin. Anal. RWA, 10 (2009), 723-744.  doi: 10.1016/j.nonrwa.2007.11.005.  Google Scholar

[38]

M. Haque and D. Greenhalgh, When a predator avoids infected prey: A model-based theoretical study, Math. Med. Biol., 27 (2010), 75-94.  doi: 10.1093/imammb/dqp007.  Google Scholar

[39]

M. HaqueJ. Zhen and E. Venturino, An ecoepidemiological predator-prey model with standard disease incidence, Math. Meth. Appl. Sci., 32 (2009), 875-898.  doi: 10.1002/mma.1071.  Google Scholar

[40]

H. W. HethcoteW. WangL. Han and Z. Ma, A predator-prey model with infected prey, Theor. Pop. Biol., 66 (2004), 259-268.  doi: 10.1016/j.tpb.2004.06.010.  Google Scholar

[41]

H. W. HethcoteW. Wang and Y. Li, Species coexistence and periodicity in host-host-pathogen models, J. Math. Biol., 51 (2005), 629-660.  doi: 10.1007/s00285-005-0335-5.  Google Scholar

[42]

F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, J. Biol. Dyn., 4 (2010), 86-101.  doi: 10.1080/17513750903026429.  Google Scholar

[43]

S. Hsu, S. Ruan and T.-H. Yang, Mathematical modelling of prey-predator interaction with disease in prey, Int. J. Comp. Sci. Math., 7. Google Scholar

[44]

P. J. HudsonA. P. Dobson and D. Newborn, Do parasites make prey vulnerable to predation? red grouse and parasites, J. Animal Ecol., 61 (1992), 681-692.  doi: 10.2307/5623.  Google Scholar

[45]

A. D. Jassby and T. Platt, Mathematical formulation of the relationship between photosynthesis and light for phytoplankton, Limnology and oceanography, 21 (1976), 540-547.  doi: 10.4319/lo.1976.21.4.0540.  Google Scholar

[46]

Q. J. A. KhanM. A. Al-Lawatia and F. Al-Kharousi, Predator-prey harvesting model with fatal disease in prey, Math. Meth. in App. Sci., 39 (2016), 2647-2658.  doi: 10.1002/mma.3718.  Google Scholar

[47]

H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Pattern in Ecology and Epidemiology. Theory, Models, and Simulation, Chapman & Hall/CRC, Boca Raton, FL, 2008.  Google Scholar

[48]

D. Mukherjee, Persistence aspect of a predator-prey model with disease in the prey, Differ. Equ. Dyn. Syst., 24 (2016), 173-188.  doi: 10.1007/s12591-014-0213-y.  Google Scholar

[49]

L. J. RachowiczJ. M. HeroR. A. AlfordJ. W. TaylorV. T. VredenburgJ. A. T. MorganJ. P. Collins and C. J. Briggs, The novel and endemic pathogen hypotheses: competing explanations for the origin of emerging infectious diseases of wildlife, Conserv. Biol., 19 (2005), 1441-1448.  doi: 10.1111/j.1523-1739.2005.00255.x.  Google Scholar

[50]

S. G. Ruan and H. I. Freedman, Persistence in three-species food chain models with group defense, Math. Biosci., 107 (1991), 111-125.  doi: 10.1016/0025-5564(91)90074-S.  Google Scholar

[51]

S. SarwardiM. Haque and E. Venturino, A Leslie-Gower Holling-type Ⅱ ecoepidemic model, J. of App. Math. Comp., 35 (2011), 263-280.  doi: 10.1007/s12190-009-0355-1.  Google Scholar

[52]

S. SarwardiM. Haque and E. Venturino, Global stability and persistence in LG-Holling type Ⅱ diseased predator ecosystems, J. Biol. Phys., 37 (2011), 91-106.  doi: 10.1007/s10867-010-9201-9.  Google Scholar

[53]

G. Seo and G. S. K. Wolkowicz, Existence of multiple limit cycles in a predator-prey model with arctan(ax) as functional response, Commun. Math. Anal., 18 (2015), 64-68.   Google Scholar

[54]

G. Seo and G. S. K. Wolkowicz, Sensitivity of the dynamics of the general rosenzweig-macarthur model to the mathematical form of the functional response: A bifurcation theory approach, J. Math. Biol, 76 (2018), 1873-1906.  doi: 10.1007/s00285-017-1201-y.  Google Scholar

[55]

B. K. SinghJ. Chattopadhyay and S. Sinha, The role of virus infection in a simple phytoplankton zooplankton system, J. Theoret. Biol., 231 (2004), 153-166.  doi: 10.1016/j.jtbi.2004.06.010.  Google Scholar

[56]

J.-J. E. Slotine and W. Li et al., Applied Nonlinear Control, vol. 199, Prentice hall Englewood Cliffs, NJ, 1991. Google Scholar

[57]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer. Math. Soc, Providence, 2011. doi: 10.1090/gsm/118.  Google Scholar

[58]

S. A. Temple, Do predators always capture substandard individuals disproportionately from prey population?, Ecology, 68 (1987), 669-674.  doi: 10.2307/1938472.  Google Scholar

[59]

H. R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111 (1992), 99-130.  doi: 10.1016/0025-5564(92)90081-7.  Google Scholar

[60] H. R. Thieme, Mathematical Population Biology, Princeton University Press, Princeton, 2003.   Google Scholar
[61]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[62]

H. R. ThiemeT. DhirasakdanonZ. Han and R. Trevino, Species decline and extinction: Synergy of infectious disease and Allee effect?, J. Biol. Dyn., 3 (2009), 305-323.  doi: 10.1080/17513750802376313.  Google Scholar

[63]

P. K. TiwariS. K. SasmalA. ShaE. Venturino and J. Chattopadhyay, Effect of diseases on symbiotic systems, BioSystems, 159 (2017), 36-50.  doi: 10.1016/j.biosystems.2017.07.001.  Google Scholar

[64]

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

[65]

E. Venturino, Ecoepidemiology: A more comprehensive view of population interactions, Math. Model. Nat. Phenom., 11 (2016), 49-90.  doi: 10.1051/mmnp/201611104.  Google Scholar

[66]

E. Venturino, O. Arino, D. Axelrod, M. Kimmel, M. Langlais and eds., Epidemics in predatorprey models: disease in the prey, Math. Pop. Dyn.: Anal. of Hetero., Vol. One: Theory of Epidemics, 92 (1995), 381-393. Google Scholar

[67]

M. Q. WilberR. A. KnappM. Toothman and C. J. Briggs, Resistance, tolerance and environmental transmission dynamics determine host extinction risk in a load-dependent amphibian disease, Ecol. Lett., 20 (2017), 1169-1181.  doi: 10.1111/ele.12814.  Google Scholar

[68]

Y. Xiao and L. Chen, Analysis of a three species eco-epidemiological model, J. Math. Anal. Appl., 258 (2001), 733-754.  doi: 10.1006/jmaa.2001.7514.  Google Scholar

[69]

Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171 (2001), 59-82.  doi: 10.1016/S0025-5564(01)00049-9.  Google Scholar

[70]

Y. Xiao and L. Chen, A ratio-dependent predator-prey model with disease in the prey, App. Math. Comp., 131 (2002), 397-414.  doi: 10.1016/S0096-3003(01)00156-4.  Google Scholar

[71]

P. YongzhenL. Shuping and L. Changgua, Effect of delay on a predator-prey model with parasitic infection, Nonlinear Dyn., 63 (2011), 311-321.  doi: 10.1007/s11071-010-9805-4.  Google Scholar

[72]

E. F. ZipkinG. V. DiRenzoJ. M. RayS. Rossman and K. R. Lips, Tropical snake diversity collapses after widespread amphibian loss, Science, 367 (2020), 814-816.  doi: 10.1126/science.aay5733.  Google Scholar

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

[2]

O. ArinoA. El abdllaouiJ. 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.  Google Scholar

[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.   Google Scholar

[4]

N. BairagiP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. P. BeraA. 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]

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.  Google Scholar

[11]

J. ChattopadhyayS. 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.  Google Scholar

[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

[14] J. P. CollinsM. L. Crump and T. E. Lovejoy III, Extinction in our Times: Global Amphibian Decline, Oxford University Press, 2009.   Google Scholar
[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 AssisE. MassadR. A. de AssisS. 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. DochertyC. U. MeteyerJ. WangJ. MaoS. 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. FarrellJ. P. CollinsA. 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. FarrellJ. P. CollinsA. 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. GimmelliB. 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. GrayD. 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. GreenK. 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]

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.   Google Scholar

[32]

A. L. GreerC. 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.  Google Scholar

[33]

K. P. HadelerK. 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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

L. HanZ. 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.  Google Scholar

[37]

L. Han and A. Pugliese, Epidemics in two competing species, Nonlin. Anal. RWA, 10 (2009), 723-744.  doi: 10.1016/j.nonrwa.2007.11.005.  Google Scholar

[38]

M. Haque and D. Greenhalgh, When a predator avoids infected prey: A model-based theoretical study, Math. Med. Biol., 27 (2010), 75-94.  doi: 10.1093/imammb/dqp007.  Google Scholar

[39]

M. HaqueJ. Zhen and E. Venturino, An ecoepidemiological predator-prey model with standard disease incidence, Math. Meth. Appl. Sci., 32 (2009), 875-898.  doi: 10.1002/mma.1071.  Google Scholar

[40]

H. W. HethcoteW. WangL. Han and Z. Ma, A predator-prey model with infected prey, Theor. Pop. Biol., 66 (2004), 259-268.  doi: 10.1016/j.tpb.2004.06.010.  Google Scholar

[41]

H. W. HethcoteW. Wang and Y. Li, Species coexistence and periodicity in host-host-pathogen models, J. Math. Biol., 51 (2005), 629-660.  doi: 10.1007/s00285-005-0335-5.  Google Scholar

[42]

F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, J. Biol. Dyn., 4 (2010), 86-101.  doi: 10.1080/17513750903026429.  Google Scholar

[43]

S. Hsu, S. Ruan and T.-H. Yang, Mathematical modelling of prey-predator interaction with disease in prey, Int. J. Comp. Sci. Math., 7. Google Scholar

[44]

P. J. HudsonA. P. Dobson and D. Newborn, Do parasites make prey vulnerable to predation? red grouse and parasites, J. Animal Ecol., 61 (1992), 681-692.  doi: 10.2307/5623.  Google Scholar

[45]

A. D. Jassby and T. Platt, Mathematical formulation of the relationship between photosynthesis and light for phytoplankton, Limnology and oceanography, 21 (1976), 540-547.  doi: 10.4319/lo.1976.21.4.0540.  Google Scholar

[46]

Q. J. A. KhanM. A. Al-Lawatia and F. Al-Kharousi, Predator-prey harvesting model with fatal disease in prey, Math. Meth. in App. Sci., 39 (2016), 2647-2658.  doi: 10.1002/mma.3718.  Google Scholar

[47]

H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Pattern in Ecology and Epidemiology. Theory, Models, and Simulation, Chapman & Hall/CRC, Boca Raton, FL, 2008.  Google Scholar

[48]

D. Mukherjee, Persistence aspect of a predator-prey model with disease in the prey, Differ. Equ. Dyn. Syst., 24 (2016), 173-188.  doi: 10.1007/s12591-014-0213-y.  Google Scholar

[49]

L. J. RachowiczJ. M. HeroR. A. AlfordJ. W. TaylorV. T. VredenburgJ. A. T. MorganJ. P. Collins and C. J. Briggs, The novel and endemic pathogen hypotheses: competing explanations for the origin of emerging infectious diseases of wildlife, Conserv. Biol., 19 (2005), 1441-1448.  doi: 10.1111/j.1523-1739.2005.00255.x.  Google Scholar

[50]

S. G. Ruan and H. I. Freedman, Persistence in three-species food chain models with group defense, Math. Biosci., 107 (1991), 111-125.  doi: 10.1016/0025-5564(91)90074-S.  Google Scholar

[51]

S. SarwardiM. Haque and E. Venturino, A Leslie-Gower Holling-type Ⅱ ecoepidemic model, J. of App. Math. Comp., 35 (2011), 263-280.  doi: 10.1007/s12190-009-0355-1.  Google Scholar

[52]

S. SarwardiM. Haque and E. Venturino, Global stability and persistence in LG-Holling type Ⅱ diseased predator ecosystems, J. Biol. Phys., 37 (2011), 91-106.  doi: 10.1007/s10867-010-9201-9.  Google Scholar

[53]

G. Seo and G. S. K. Wolkowicz, Existence of multiple limit cycles in a predator-prey model with arctan(ax) as functional response, Commun. Math. Anal., 18 (2015), 64-68.   Google Scholar

[54]

G. Seo and G. S. K. Wolkowicz, Sensitivity of the dynamics of the general rosenzweig-macarthur model to the mathematical form of the functional response: A bifurcation theory approach, J. Math. Biol, 76 (2018), 1873-1906.  doi: 10.1007/s00285-017-1201-y.  Google Scholar

[55]

B. K. SinghJ. Chattopadhyay and S. Sinha, The role of virus infection in a simple phytoplankton zooplankton system, J. Theoret. Biol., 231 (2004), 153-166.  doi: 10.1016/j.jtbi.2004.06.010.  Google Scholar

[56]

J.-J. E. Slotine and W. Li et al., Applied Nonlinear Control, vol. 199, Prentice hall Englewood Cliffs, NJ, 1991. Google Scholar

[57]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer. Math. Soc, Providence, 2011. doi: 10.1090/gsm/118.  Google Scholar

[58]

S. A. Temple, Do predators always capture substandard individuals disproportionately from prey population?, Ecology, 68 (1987), 669-674.  doi: 10.2307/1938472.  Google Scholar

[59]

H. R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111 (1992), 99-130.  doi: 10.1016/0025-5564(92)90081-7.  Google Scholar

[60] H. R. Thieme, Mathematical Population Biology, Princeton University Press, Princeton, 2003.   Google Scholar
[61]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[62]

H. R. ThiemeT. DhirasakdanonZ. Han and R. Trevino, Species decline and extinction: Synergy of infectious disease and Allee effect?, J. Biol. Dyn., 3 (2009), 305-323.  doi: 10.1080/17513750802376313.  Google Scholar

[63]

P. K. TiwariS. K. SasmalA. ShaE. Venturino and J. Chattopadhyay, Effect of diseases on symbiotic systems, BioSystems, 159 (2017), 36-50.  doi: 10.1016/j.biosystems.2017.07.001.  Google Scholar

[64]

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

[65]

E. Venturino, Ecoepidemiology: A more comprehensive view of population interactions, Math. Model. Nat. Phenom., 11 (2016), 49-90.  doi: 10.1051/mmnp/201611104.  Google Scholar

[66]

E. Venturino, O. Arino, D. Axelrod, M. Kimmel, M. Langlais and eds., Epidemics in predatorprey models: disease in the prey, Math. Pop. Dyn.: Anal. of Hetero., Vol. One: Theory of Epidemics, 92 (1995), 381-393. Google Scholar

[67]

M. Q. WilberR. A. KnappM. Toothman and C. J. Briggs, Resistance, tolerance and environmental transmission dynamics determine host extinction risk in a load-dependent amphibian disease, Ecol. Lett., 20 (2017), 1169-1181.  doi: 10.1111/ele.12814.  Google Scholar

[68]

Y. Xiao and L. Chen, Analysis of a three species eco-epidemiological model, J. Math. Anal. Appl., 258 (2001), 733-754.  doi: 10.1006/jmaa.2001.7514.  Google Scholar

[69]

Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171 (2001), 59-82.  doi: 10.1016/S0025-5564(01)00049-9.  Google Scholar

[70]

Y. Xiao and L. Chen, A ratio-dependent predator-prey model with disease in the prey, App. Math. Comp., 131 (2002), 397-414.  doi: 10.1016/S0096-3003(01)00156-4.  Google Scholar

[71]

P. YongzhenL. Shuping and L. Changgua, Effect of delay on a predator-prey model with parasitic infection, Nonlinear Dyn., 63 (2011), 311-321.  doi: 10.1007/s11071-010-9805-4.  Google Scholar

[72]

E. F. ZipkinG. V. DiRenzoJ. M. RayS. Rossman and K. R. Lips, Tropical snake diversity collapses after widespread amphibian loss, Science, 367 (2020), 814-816.  doi: 10.1126/science.aay5733.  Google Scholar

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}} $
$ ^{{\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}} $
[1]

Luca Gerardo-Giorda, Pierre Magal, Shigui Ruan, Ousmane Seydi, Glenn Webb. Preface: Population dynamics in epidemiology and ecology. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : i-ii. doi: 10.3934/dcdsb.2020125

[2]

Pikkala Vijaya Laxmi, Seleshi Demie. Performance analysis of renewal input $(a,c,b)$ policy queue with multiple working vacations and change over times. Journal of Industrial & Management Optimization, 2014, 10 (3) : 839-857. doi: 10.3934/jimo.2014.10.839

[3]

Graeme Wake, Anthony Pleasants, Alan Beedle, Peter Gluckman. A model for phenotype change in a stochastic framework. Mathematical Biosciences & Engineering, 2010, 7 (3) : 719-728. doi: 10.3934/mbe.2010.7.719

[4]

Diana M. Thomas, Ashley Ciesla, James A. Levine, John G. Stevens, Corby K. Martin. A mathematical model of weight change with adaptation. Mathematical Biosciences & Engineering, 2009, 6 (4) : 873-887. doi: 10.3934/mbe.2009.6.873

[5]

M. Núñez-López, J. X. Velasco-Hernández, P. A. Marquet. The dynamics of technological change under constraints: Adopters and resources. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3299-3317. doi: 10.3934/dcdsb.2014.19.3299

[6]

Hans-Christoph Grunau, Guido Sweers. A clamped plate with a uniform weight may change sign. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 761-766. doi: 10.3934/dcdss.2014.7.761

[7]

Hung-Wen Kuo. Effect of abrupt change of the wall temperature in the kinetic theory. Kinetic & Related Models, 2019, 12 (4) : 765-789. doi: 10.3934/krm.2019030

[8]

L. Aguirre, P. Seibert. Types of change of stability and corresponding types of bifurcations. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 741-752. doi: 10.3934/dcds.1999.5.741

[9]

Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525

[10]

Yuri Nechepurenko, Michael Khristichenko, Dmitry Grebennikov, Gennady Bocharov. Bistability analysis of virus infection models with time delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2385-2401. doi: 10.3934/dcdss.2020166

[11]

Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207

[12]

Jianquan Li, Xiaoqin Wang, Xiaolin Lin. Impact of behavioral change on the epidemic characteristics of an epidemic model without vital dynamics. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1425-1434. doi: 10.3934/mbe.2018065

[13]

Xuchen Lin, Ting-Jie Lu, Xia Chen. Total factor productivity growth and technological change in the telecommunications industry. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 795-809. doi: 10.3934/dcdss.2019053

[14]

José Luiz Boldrini, Gabriela Planas. A tridimensional phase-field model with convection for phase change of an alloy. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 429-450. doi: 10.3934/dcds.2005.13.429

[15]

Martha G. Alatriste-Contreras, Juan Gabriel Brida, Martin Puchet Anyul. Structural change and economic dynamics: Rethinking from the complexity approach. Journal of Dynamics & Games, 2019, 6 (2) : 87-106. doi: 10.3934/jdg.2019007

[16]

Miguel Atencia, Esther García-Garaluz, Gonzalo Joya. The ratio of hidden HIV infection in Cuba. Mathematical Biosciences & Engineering, 2013, 10 (4) : 959-977. doi: 10.3934/mbe.2013.10.959

[17]

Kokum R. De Silva, Shigetoshi Eda, Suzanne Lenhart. Modeling environmental transmission of MAP infection in dairy cows. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1001-1017. doi: 10.3934/mbe.2017052

[18]

M. M. Ali, L. Masinga. A nonlinear optimization model for optimal order quantities with stochastic demand rate and price change. Journal of Industrial & Management Optimization, 2007, 3 (1) : 139-154. doi: 10.3934/jimo.2007.3.139

[19]

Andrei V. Dmitruk, Nikolai P. Osmolovskii. Proof of the maximum principle for a problem with state constraints by the v-change of time variable. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2189-2204. doi: 10.3934/dcdsb.2019090

[20]

Reimund Rautmann. Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1101-1109. doi: 10.3934/dcdss.2014.7.1101

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (183)
  • HTML views (163)
  • Cited by (0)

Other articles
by authors

[Back to Top]