Predator - prey/host - parasite: a fragile ecoepidemic system under homogeneous infection incidence. (English) Zbl 1466.92221
Summary: 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.
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.
MSC:
| 92D40 | Ecology |
| 92D25 | Population dynamics (general) |
| 92D30 | Epidemiology |
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
Keywords:
coexistence; multiple interior equilibria; bistability; homogeneous incidence; infection ratio; environmental change; climate change; ecology; epidemiologyReferences:
| [1] | F. Abbona; 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, 2269-2280 (2018) · Zbl 1391.92045 · doi:10.1002/mma.4209 |
| [2] | O. Arino; A. El abdllaoui; J. Mikram; J. Chattopadhyay, Infection in prey population may act as a biological control in ratio-dependent predator-prey models, Nonlinearity, 17, 1101-1116 (2004) · Zbl 1062.34050 · doi:10.1088/0951-7715/17/3/018 |
| [3] | B. S. Attili; 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, 33-40 (2006) · Zbl 1121.92064 |
| [4] | N. Bairagi; P. K. Roy; J. Chattopadhyay, Role of infection on the stability of a predator-prey system with several response functionsa comparative study, J. Theo. Biol., 248, 10-25 (2007) · Zbl 1451.92274 · 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. · Zbl 1515.92067 |
| [6] | E. Beltrami; T. O. Carroll, Modeling the role of viral disease in recurrent phytoplankton bloom, Journal of Mathematical Biology, 32, 857-863 (1994) · Zbl 0825.92122 · doi:10.1007/BF00168802 |
| [7] | S. P. Bera; A. Maiti; G. P. Samanta, Prey-predator model with infection in both prey and predator, Filomat, 29, 1753-1767 (2015) · Zbl 1403.92235 · doi:10.2298/FIL1508753B |
| [8] | E. Cagliero; E. Venturino, Ecoepidemics with infected prey in herd defence: The harmless and toxic cases, Int. J. Comp. Math., 93, 108-127 (2016) · Zbl 1357.92068 · doi:10.1080/00207160.2014.988614 |
| [9] | J. Chattopadhyay; O. Arino, A predator-prey model with disease in the prey, Nonlin. Anal., 36, 747-766 (1999) · Zbl 0922.34036 · doi:10.1016/S0362-546X(98)00126-6 |
| [10] | J. Chattopadhyay; S. Pal, Viral infection on phytoplankton zooplankton system a mathematical model, Ecological Modeling, 151, 15-28 (2002) · doi:10.1016/S0304-3800(01)00415-X |
| [11] | J. Chattopadhyay; S. Pal; A. El Abdllaoui, Classical predator-prey system with infection of prey populationa mathematical model, Math. Meth. in the App. Sci., 26, 1211-1222 (2003) · Zbl 1044.34001 · doi:10.1002/mma.414 |
| [12] | Y. Chen; Y. Wen, Impact on the predator population while lethal disease spreads in the prey, Math. Meth. in App. Sci., 39, 2883-2895 (2016) · Zbl 1343.92397 · 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, 93-99 (2010) · doi:10.3354/dao02307 |
| [14] | 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. · Zbl 1328.92059 |
| [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. · Zbl 1382.92234 · doi:10.1002/mma.4568 |
| [16] | L. M. E. de Assis; E. Massad; R. A. de Assis; S. R. Martorano; E. Venturino, A mathematical model for bovine tuberculosis among buffaloes and lions in the kruger national park, Mathematical Methods in the Applied Sciences, 41, 525-543 (2018) · Zbl 0559.47040 · doi:10.1002/mma.4568 |
| [17] | K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. · Zbl 1205.92063 · doi:10.1051/mmnp/20105606 |
| [18] | T. Dhirasakdanon; H. Thieme, Stability of the endemic coexistence equilibrium for one host and two parasites, Mathematical Modelling of Natural Phenomena, 5, 109-138 (2010) · Zbl 1205.92063 · doi:10.1051/mmnp/20105606 |
| [19] | D. E. Docherty; C. U. Meteyer; J. Wang; J. Mao; S. T. Case; V. G. Chinchar, Diagnostic and molecular evaluation of three iridovirus-associated salamander mortality events, J. Wildl. Dis., 39, 556-566 (2003) · 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. · Zbl 1404.92181 · doi:10.1007/s00285-018-1249-3 |
| [21] | A. P. Farrell; J. P. Collins; A. L. Greer; H. R. Thieme, Do fatal infectious diseases eradicate host species?, J. Math. Biol., 77, 2103-2164 (2018) · Zbl 1396.92082 · doi:10.1007/s00285-018-1249-3 |
| [22] | A. P. Farrell; J. P. Collins; A. L. Greer; H. R. Thieme, Times from infection to disease-induced death and their influence on final population sizes after epidemic outbreaks, Bull. Math. Biol., 80, 1937-1961 (2018) · Zbl 1273.34052 · doi:10.1007/s11538-018-0446-y |
| [23] | A. Friedman; A.-A. Yakubu, Host demographic Allee effect, fatal disease, and migration: Persistence or extinction, SIAM J. Appl. Math., 72, 1644-1666 (2012) · Zbl 1277.92033 · doi:10.1137/120861382 |
| [24] | J. Gani; R. J. Swift, Prey-predator models with infected prey and predators, Disc. and Cont. Dyn. Sys., 33, 5059-5066 (2013) · Zbl 1277.92033 · doi:10.3934/dcds.2013.33.5059 |
| [25] | W. M. Getz; J. Pickering, Epidemic models: Thresholds and population regulation, Am. Nat., 121, 892-898 (1983) · doi:10.1086/284112 |
| [26] | M. Ghosh; X.-Z. Li, Mathematical modelling of prey-predator interaction with disease in prey, Int. J. Comp. Sci. Math., 7, 443-458 (2016) · Zbl 1453.92244 · doi:10.1504/IJCSM.2016.080075 |
| [27] | G. Gimmelli; B. W. Kooi; E. Venturino, Ecoepidemic models with prey group defense and feeding saturation, Ecol. Complex., 22, 50-58 (2015) · doi:10.1016/j.ecocom.2015.02.004 |
| [28] | M. J. Gray and V. G. Chinchar, Ranaviruses: Lethal Pathogens of Ectothermic Vertebrates, Springer, 2015. · doi:10.3354/dao02138 |
| [29] | M. J. Gray; D. L. Miller; J. T. Hoverman, Ecology and pathology of amphibian ranaviruses, Dis. Aquat. Organ., 87, 243-266 (2009) · doi:10.3354/dao02138 |
| [30] | D. E. Green; K. A. Converse; A. K. Schrader, Epizootiology of sixty-four amphibian morbidity and mortality events in the USA, 1996-2001, Ann. N. Y. Acad. Sci., 969, 323-339 (2002) · doi:10.1111/j.1749-6632.2002.tb04400.x |
| [31] | D. Greenhalgh; 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, 79-101 (1995) · doi:10.1111/j.1600-0706.2008.16783.x |
| [32] | A. L. Greer; C. J. Briggs; J. P. Collins, Testing a key assumption of host-pathogen theory: Density and disease transmission, Oikos, 117, 1667-1673 (2008) · Zbl 1348.92152 · doi:10.1111/j.1600-0706.2008.16783.x |
| [33] | K. P. Hadeler; K. Dietz; M. Safan, Case fatality models for epidemics in growing populations, Math. Biosci., 281, 120-127 (2016) · Zbl 0716.92021 · doi:10.1016/j.mbs.2016.09.007 |
| [34] | K. P. Hadeler; H. I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol., 27, 609-631 (1989) · Zbl 0433.34003 · doi:10.1007/BF00276947 |
| [35] | J. K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Company, Inc., 1980. · Zbl 0999.92032 · doi:10.1016/S0895-7177(01)00104-2 |
| [36] | L. Han; Z. Ma; H. W. Hethcote, Four predator prey models with infectious diseases, Math. Comp. Modeling, 34, 849-858 (2001) · Zbl 1167.34358 · doi:10.1016/S0895-7177(01)00104-2 |
| [37] | L. Han; A. Pugliese, Epidemics in two competing species, Nonlin. Anal. RWA, 10, 723-744 (2009) · Zbl 1184.92052 · doi:10.1016/j.nonrwa.2007.11.005 |
| [38] | M. Haque; D. Greenhalgh, When a predator avoids infected prey: A model-based theoretical study, Math. Med. Biol., 27, 75-94 (2010) · Zbl 1158.92035 · doi:10.1093/imammb/dqp007 |
| [39] | M. Haque; J. Zhen; E. Venturino, An ecoepidemiological predator-prey model with standard disease incidence, Math. Meth. Appl. Sci., 32, 875-898 (2009) · Zbl 1158.92035 · doi:10.1002/mma.1071 |
| [40] | H. W. Hethcote; W. Wang; L. Han; Z. Ma, A predator-prey model with infected prey, Theor. Pop. Biol., 66, 259-268 (2004) · Zbl 1077.92044 · doi:10.1016/j.tpb.2004.06.010 |
| [41] | H. W. Hethcote; W. Wang; Y. Li, Species coexistence and periodicity in host-host-pathogen models, J. Math. Biol., 51, 629-660 (2005) · Zbl 1315.92060 · doi:10.1007/s00285-005-0335-5 |
| [42] | F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, J. Biol. Dyn., 4, 86-101 (2010) · Zbl 1315.92060 · doi:10.1080/17513750903026429 |
| [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. · Zbl 1453.92244 · doi:10.2307/5623 |
| [44] | P. J. Hudson; A. P. Dobson; D. Newborn, Do parasites make prey vulnerable to predation? red grouse and parasites, J. Animal Ecol., 61, 681-692 (1992) · doi:10.2307/5623 |
| [45] | A. D. Jassby; T. Platt, Mathematical formulation of the relationship between photosynthesis and light for phytoplankton, Limnology and oceanography, 21, 540-547 (1976) · Zbl 1341.92059 · doi:10.4319/lo.1976.21.4.0540 |
| [46] | Q. J. A. Khan; M. A. Al-Lawatia; F. Al-Kharousi, Predator-prey harvesting model with fatal disease in prey, Math. Meth. in App. Sci., 39, 2647-2658 (2016) · Zbl 1298.92004 · doi:10.1002/mma.3718 |
| [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. · Zbl 1358.92080 · doi:10.1007/s12591-014-0213-y |
| [48] | D. Mukherjee, Persistence aspect of a predator-prey model with disease in the prey, Differ. Equ. Dyn. Syst., 24, 173-188 (2016) · Zbl 1358.92080 · doi:10.1007/s12591-014-0213-y |
| [49] | L. J. Rachowicz; J. M. Hero; R. A. Alford; J. W. Taylor; V. T. Vredenburg; J. A. T. Morgan; J. P. Collins; C. J. Briggs, The novel and endemic pathogen hypotheses: competing explanations for the origin of emerging infectious diseases of wildlife, Conserv. Biol., 19, 1441-1448 (2005) · Zbl 0752.92027 · doi:10.1111/j.1523-1739.2005.00255.x |
| [50] | S. G. Ruan; H. I. Freedman, Persistence in three-species food chain models with group defense, Math. Biosci., 107, 111-125 (1991) · Zbl 1209.92060 · doi:10.1016/0025-5564(91)90074-S |
| [51] | S. Sarwardi; M. Haque; E. Venturino, A Leslie-Gower Holling-type Ⅱ ecoepidemic model, J. of App. Math. Comp., 35, 263-280 (2011) · Zbl 1209.92060 · doi:10.1007/s12190-009-0355-1 |
| [52] | S. Sarwardi; M. Haque; E. Venturino, Global stability and persistence in LG-Holling type Ⅱ diseased predator ecosystems, J. Biol. Phys., 37, 91-106 (2011) · Zbl 1320.92073 · doi:10.1007/s10867-010-9201-9 |
| [53] | G. Seo; G. S. K. Wolkowicz, Existence of multiple limit cycles in a predator-prey model with arctan(ax) as functional response, Commun. Math. Anal., 18, 64-68 (2015) · Zbl 1390.92123 · doi:10.1007/s00285-017-1201-y |
| [54] | G. Seo; 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, 1873-1906 (2018) · Zbl 1447.92475 · doi:10.1007/s00285-017-1201-y |
| [55] | B. K. Singh; J. Chattopadhyay; S. Sinha, The role of virus infection in a simple phytoplankton zooplankton system, J. Theoret. Biol., 231, 153-166 (2004) · Zbl 1447.92475 · doi:10.1016/j.jtbi.2004.06.010 |
| [56] | J.-J. E. Slotine and W. Li et al., Applied Nonlinear Control, vol. 199, Prentice hall Englewood Cliffs, NJ, 1991. · Zbl 1214.37002 |
| [57] | H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer. Math. Soc, Providence, 2011. · Zbl 1214.37002 · doi:10.2307/1938472 |
| [58] | S. A. Temple, Do predators always capture substandard individuals disproportionately from prey population?, Ecology, 68, 669-674 (1987) · Zbl 0782.92018 · doi:10.2307/1938472 |
| [59] | H. R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111, 99-130 (1992) · Zbl 0761.34039 · doi:10.1016/0025-5564(92)90081-7 |
| [60] | H. R. Thieme; T. Dhirasakdanon; Z. Han; R. Trevino, Species decline and extinction: Synergy of infectious disease and Allee effect?, J. Biol. Dyn., 3, 305-323 (2009) · Zbl 1342.92206 · doi:10.1080/17513750802376313 |
| [61] | H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30, 755-763 (1992) · Zbl 0761.34039 · doi:10.1007/BF00173267 |
| [62] | H. R. Thieme; T. Dhirasakdanon; Z. Han; R. Trevino, Species decline and extinction: Synergy of infectious disease and Allee effect?, J. Biol. Dyn., 3, 305-323 (2009) · Zbl 0799.92017 · doi:10.1080/17513750802376313 |
| [63] | P. K. Tiwari; S. K. Sasmal; A. Sha; E. Venturino; J. Chattopadhyay, Effect of diseases on symbiotic systems, BioSystems, 159, 36-50 (2017) · Zbl 1384.92060 · doi:10.1016/j.biosystems.2017.07.001 |
| [64] | E. Venturino, The influence of diseases on Lotka-Volterra systems, Rocky Mt. J. Math., 24, 381-402 (1994) · Zbl 0799.92017 · doi:10.1216/rmjm/1181072471 |
| [65] | E. Venturino, Ecoepidemiology: A more comprehensive view of population interactions, Math. Model. Nat. Phenom., 11, 49-90 (2016) · Zbl 1384.92060 · doi:10.1051/mmnp/201611104 |
| [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. · Zbl 0967.92017 · doi:10.1006/jmaa.2001.7514 |
| [67] | M. Q. Wilber; R. A. Knapp; M. Toothman; C. J. Briggs, Resistance, tolerance and environmental transmission dynamics determine host extinction risk in a load-dependent amphibian disease, Ecol. Lett., 20, 1169-1181 (2017) · Zbl 0978.92031 · doi:10.1111/ele.12814 |
| [68] | Y. Xiao; L. Chen, Analysis of a three species eco-epidemiological model, J. Math. Anal. Appl., 258, 733-754 (2001) · Zbl 1024.92017 · doi:10.1006/jmaa.2001.7514 |
| [69] | Y. Xiao; L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171, 59-82 (2001) · Zbl 0978.92031 · doi:10.1016/S0025-5564(01)00049-9 |
| [70] | Y. Xiao; L. Chen, A ratio-dependent predator-prey model with disease in the prey, App. Math. Comp., 131, 397-414 (2002) · Zbl 1024.92017 · doi:10.1016/S0096-3003(01)00156-4 |
| [71] | P. Yongzhen; L. Shuping; L. Changgua, Effect of delay on a predator-prey model with parasitic infection, Nonlinear Dyn., 63, 311-321 (2011) · doi:10.1007/s11071-010-9805-4 |
| [72] | E. F. Zipkin; G. V. DiRenzo; J. M. Ray; S. Rossman; K. R. Lips, Tropical snake diversity collapses after widespread amphibian loss, Science, 367, 814-816 (2020) · doi:10.1126/science.aay5733 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
