International Journal of Dynamics and Control

, Volume 7, Issue 4, pp 1195–1212 | Cite as

The effect of additional food in Holling Tanner type models

  • Aladeen Basheer
  • Emmanuel Quansah
  • Rana D. ParshadEmail author


Biological control, the use of predators and pathogens to control target pests, is a promising alternative to chemical control. It is hypothesized that the introduced predators efficacy can be boosted by providing them with an additional food source. The current literature primarily considers models of additional food in predator–pest systems for symmetric predator functional responses—that is when the functional and numerical responses are of the same form. The purpose of the current manuscript is to show that providing predators with additional food in models where their functional response is not symmetric, such as Holling–Tanner models, is also effective in pest eradication. Results on stability, cyclicity and Turing instability in such a class of models is investigated. Our results have large scale implications for the effective design of biological control methods involving additional food.


Additional food Stability Turing instability Biological invasions Biological control 

Mathematics Subject Classification

Primary 34C11 34D05 35K57 60H10 Secondary 92D25 92D40 



RP would like to acknowledge partial support from the National Science Foundation via DMS 1715377 and DMS 1839993.


  1. 1.
    Pimentel D, Zuniga R, Morrison D (2005) Update on the environmental and economic costs associated with alien-invasive species in the United States. Ecol Econ 52(3):273–288Google Scholar
  2. 2.
    Lewis MA, Petrovskii SV, Potts JR (2016) The mathematics behind biological invasions, vol 44. Springer, BerlinzbMATHGoogle Scholar
  3. 3.
    Paini DR, Sheppard AW, Cook DC, De Barro PJ, Worner SP, Thomas MB (2016) Global threat to agriculture from invasive species. Proc Natl Acad Sci 113(27):7575–7579Google Scholar
  4. 4.
    Jongejans E, Shea K, Skarpaas O, Kelly D, Ellner SP (2011) Importance of individual and environmental variation for invasive species spread: a spatial integral projection model. Ecology 92(1):86–97Google Scholar
  5. 5.
    Parshad RD, Quansah E, Black K, Beauregard M (2016) Biological control via “ecological” damping: an approach that attenuates non-target effects. Math Biosci 273:23–44MathSciNetzbMATHGoogle Scholar
  6. 6.
    David P, Michael B (2014) Environmental and economic costs of the application of pesticides primarily in the United States. Integrated pest management. Springer, Dordrecht, pp 47–71Google Scholar
  7. 7.
    Van Driesche RS, Bellows TS (1996) Biological control. Kluwer Academic Publishers, Massachusetts Google Scholar
  8. 8.
    Czaja K, Góralczyk K, Struciński P, Hernik A, Korcz W, Minorczyk M, Łyczewska M, Ludwicki JK (2015) Biopesticides—towards increased consumer safety in the European Union. Pest Manag Sci 71(1):3–6Google Scholar
  9. 9.
    Bampfylde CJ, Lewis MA (2007) Biological control through intraguild predation: case studies in pest control, invasive species and range expansion. Bull Math Biol 69(3):1031–1066MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kang Y, Bai D, Tapia L, Bateman H (2017) Dynamical effects of biocontrol on the ecosystem: benefits or harm? Appl Math Model 51:361–385MathSciNetGoogle Scholar
  11. 11.
    Parshad RD, Upadhyay RK, Mishra S, Tiwari SK, Sharma S (2017) On the explosive instability in a three species food chain model with modified Holling type IV functional response. Math Methods Appl Sci 40(16):5707–5726MathSciNetzbMATHGoogle Scholar
  12. 12.
    Parshad RD, Bhowmick S, Quansah E, Basheer A, Upadhyay RK (2016) Predator interference effects on biological control: the paradox of the generalist predator revisited. Commun Nonlinear Sci Numer Simul 39:169–184MathSciNetGoogle Scholar
  13. 13.
    Wang X, Walton JR, Parshad RD, Storey K, Boggess M (2014) Analysis of the Trojan Y-chromosome strategy for eradication of an invasive species. J Math Biol 68(7):1731–1756MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gutierrez JB, Hurdal MK, Parshad RD, Teem JL (2012) Analysis of the Trojan Y chromosome model for eradication of exotic species in a dendritic riverine system. J Math Biol 64(1–2):319–340MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lyu J, Schofield PJ, Reaver KM, Beauregard M, Parshad RD (2019) A comparison of the Trojan Y chromosome strategy to harvesting models for eradication of non-native species. Nat Res Model (in revision)Google Scholar
  16. 16.
    Parshad RD, Gutierrez JB, Kouachi S (2013) Global existence and asymptotic behavior of a model for biological control of invasive species via supermale introduction. Commun Math Sci 11(4):971–992MathSciNetzbMATHGoogle Scholar
  17. 17.
    Srinivasu PDN, Prasad BSRV, Venkatesulu M (2007) Biological control through provision of additional food to predators: a theoretical study. Theor Popul Biol 72(1):111–120zbMATHGoogle Scholar
  18. 18.
    Wade MR, Zalucki MP, Wratten SD, Robinson KA (2008) Conservation biological control of arthropods using artificial food sprays: current status and future challenges. Biol Control 45(2):185–199Google Scholar
  19. 19.
    Evans EW, Swallow JG (1993) Numerical responses of natural enemies to artificial honeydew in Utah alfalfa. Environ Entomol 22(6):1392–1401Google Scholar
  20. 20.
    Canas LA, O’Neil RJ (1998) Applications of sugar solutions to maize, and the impact of natural enemies on fall armyworm. Int J Pest Manag 44(2):59–64Google Scholar
  21. 21.
    Snyder WE, Wise DH (1999) Predator interference and the establishment of generalist predator populations for biocontrol. Biol Control 15(3):283–292Google Scholar
  22. 22.
    Evans EW, Richards DR (1997) Managing the dispersal of ladybird beetles (Col.: Coccinellidae): use of artificial honeydew to manipulate spatial distributions. Entomophaga 42(1–2):93–102Google Scholar
  23. 23.
    Saunders G, Cooke B, McColl K, Shine R, Peacock T (2010) Modern approaches for the biological control of vertebrate pests: an Australian perspective. Biol Control 52(3):288–295Google Scholar
  24. 24.
    Shannon SP, Chrzanowski TH, Grover JP (2007) Prey food quality affects flagellate ingestion rates. Microb Ecol 53(1):66–73Google Scholar
  25. 25.
    Tena A, Pekas A, Cano D, Wäckers FL, Urbaneja A (2015) Sugar provisioning maximizes the biocontrol service of parasitoids. J Appl Ecol 52(3):795–804Google Scholar
  26. 26.
    Wetzel WC, Kharouba HM, Robinson M, Holyoak M, Karban R (2016) Variability in plant nutrients reduces insect herbivore performance. Nature 539(7629):425Google Scholar
  27. 27.
    Rosenheim JA, Kaya H, Ehler L, Marois JJ, Jaffee B (1995) Intraguild predation among biological-control agents: theory and practice. Biol Control 5(3):303–335Google Scholar
  28. 28.
    Put K, Bollens T, Wäckers FL, Pekas A (2012) Type and spatial distribution of food supplements impact population development and dispersal of the omnivore predator Macrolophus pygmaeus (Rambur) (Hemiptera: Miridae). Biol Control 63(2):172–180Google Scholar
  29. 29.
    Stephens AEA, Srivastava DS, Myers JH (2013) Strength in numbers? Effects of multiple natural enemy species on plant performance. Proc R Soc B Biol Sci 280(1760):20122756Google Scholar
  30. 30.
    Turchin P (2003) Complex population dynamics. A theoretical/empirical synthesis, monographs in population biology, vol 35. Princeton University Press, PrincetonzbMATHGoogle Scholar
  31. 31.
    Ulfa HM, Suryanto A, Darti I (2017) Dynamics of Leslie–Gower predator–prey model with additional food for predators. Int J Pure Appl Math 115(2):199–209Google Scholar
  32. 32.
    Srinivasu PDN, Prasad BSRV (2010) Time optimal control of an additional food provided predator–prey system with applications to pest management and biological conservation. J Math Biol 60(4):591–613MathSciNetzbMATHGoogle Scholar
  33. 33.
    Srinivasu PDN, Prasad BSRV (2011) Role of quantity of additional food to predators as a control in predator prey systems with relevance to pest management and biological conservation. Bull Math Biol 73(10):2249–2276MathSciNetzbMATHGoogle Scholar
  34. 34.
    Prasad BSRV, Banerjee M, Srinivasu PDN (2013) Dynamics of additional food provided predator–prey system with mutually interfering predators. Math Biosci 246(1):176–190MathSciNetzbMATHGoogle Scholar
  35. 35.
    Srinivasu PDN, Vamsi DKK, Aditya I (2018) Biological conservation of living systems by providing additional food supplements in the presence of inhibitory effect: a theoretical study using predator–prey models. Differ Equ Dyn Syst 26(1–3):213–246MathSciNetzbMATHGoogle Scholar
  36. 36.
    Parshad RD, Wickramsooriya S, Bailey S (2019) A remark on “Biological control through provision of additional food to predators: a theoretical study”. Theor Popul Biol [Theoretical Population Biology 72 (2007) 111–120] (in revision)Google Scholar
  37. 37.
    Chakraborty S, Tiwari PK, Sasmal SK, Biswas S, Bhattacharya S, Chattopadhyay J (2017) Interactive effects of prey refuge and additional food for predator in a diffusive predator–prey system. Appl Math Model 47:128–140MathSciNetGoogle Scholar
  38. 38.
    Rani R, Gakkhar S (2019) The impact of provision of additional food to predator in predator–prey model with combined harvesting in the presence of toxicity. J Appl Math Comput 60(1–2):673–701MathSciNetzbMATHGoogle Scholar
  39. 39.
    Ghosh J, Sahoo B, Poria S (2017) Prey–predator dynamics with prey refuge providing additional food to predator. Chaos Solitons Fractals 96:110–119MathSciNetzbMATHGoogle Scholar
  40. 40.
    Sasmal SK, Mandal DS, Chattopadhyay J (2017) Predator–pest model with allee effect and pest culling and additional food provision to the predator-application to pest control. J Biol Syst 25(02):295–326MathSciNetzbMATHGoogle Scholar
  41. 41.
    Gurubilli KK, Srinivasu PDN, Banerjee M (2017) Global dynamics of a prey–predator model with Allee effect and additional food for the predators. Int J Dyn Control 5(3):903–916MathSciNetGoogle Scholar
  42. 42.
    Sahoo B, Poria S (2014) The chaos and control of a food chain model supplying additional food to top-predator. Chaos Solitons Fractals 58:52–64MathSciNetzbMATHGoogle Scholar
  43. 43.
    Das A, Samanta GP (2018) Modeling the fear effect on a stochastic prey–predator system with additional food for the predator. J Phys A Math Theor 51(46):465601MathSciNetGoogle Scholar
  44. 44.
    Liu Q, Jiang D, Hayat T, Ahmad B (2018) Stationary distribution and extinction of a stochastic predator–prey model with additional food and nonlinear perturbation. Appl Math Comput 320:226–239MathSciNetzbMATHGoogle Scholar
  45. 45.
    Sahoo B, Poria S (2013) Disease control in a food chain model supplying alternative food. Appl Math Model 37(8):5653–5663MathSciNetzbMATHGoogle Scholar
  46. 46.
    Sahoo B (2015) Role of additional food in eco-epidemiological system with disease in the prey. Appl Math Comput 259:61–79MathSciNetzbMATHGoogle Scholar
  47. 47.
    Prasad KD, Prasad BSRV (2018) Biological pest control using cannibalistic predators and with provision of additional food: a theoretical study. Theor Ecol 11(2):191–211Google Scholar
  48. 48.
    Kuang Y (2007) Some mechanistically derived population models. Math Biosci Eng 4(4):1–11MathSciNetGoogle Scholar
  49. 49.
    Banerjee M, Banerjee S (2012) Turing instabilities and spatio-temporal chaos in ratio-dependent Holling–Tanner model. Math Biosci 236(1):64–76MathSciNetzbMATHGoogle Scholar
  50. 50.
    Basheer A, Quansah E, Bhowmick S, Parshad RD (2016) Prey cannibalism alters the dynamics of Holling–Tanner-type predator–prey models. Nonlinear Dyn 85(4):2549–2567MathSciNetzbMATHGoogle Scholar
  51. 51.
    Chen F (2005) On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay. J Comput Appl Math 180(1):33–49MathSciNetzbMATHGoogle Scholar
  52. 52.
    Strogatz SH (2018) Nonlinear dynamics and chaos with student solutions manual: with applications to physics, biology, chemistry, and engineering. CRC Press, Boca RatonGoogle Scholar
  53. 53.
    Liu W (1994) Criterion of Hopf bifurcations without using eigenvalues. J Math Anal Appl 182(1):250–256MathSciNetzbMATHGoogle Scholar
  54. 54.
    Haque M (2009) Ratio-dependent predator–prey models of interacting populations. Bull Math Biol 71(2):430–452MathSciNetzbMATHGoogle Scholar
  55. 55.
    Perko L (2013) Differential equations and dynamical systems. Springer, BerlinzbMATHGoogle Scholar
  56. 56.
    Rudnicki R (2003) Long-time behaviour of a stochastic prey–predator model. Stoch Process Appl 108(1):93–107MathSciNetzbMATHGoogle Scholar
  57. 57.
    Ji C, Jiang D, Shi N (2009) Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. J Math Anal Appl 359(2):482–498MathSciNetzbMATHGoogle Scholar
  58. 58.
    Xia P, Zheng X, Jiang D (2013) Persistence and nonpersistence of a nonautonomous stochastic mutualism system. In: Chu J (ed) Abstract and applied analysis, vol 2013. Hindawi, LondonMathSciNetzbMATHGoogle Scholar
  59. 59.
    Box GEP (1958) A note on the generation of random normal deviates. Ann Math Stat 29:610–611zbMATHGoogle Scholar
  60. 60.
    White KAJ, Gilligan CA (1998) Spatial heterogeneity in three species, plant–parasite–hyperparasite, systems. Philos Trans R Soc Lond Ser B Biol Sci 353(1368):543–557Google Scholar
  61. 61.
    Banerjee M, Banerjee S (2012) Turing instabilities and spatio-temporal chaos in ratio-dependent Holling–Tanner model. Math Biosci 236(1):64–76MathSciNetzbMATHGoogle Scholar
  62. 62.
    Fasani S, Rinaldi S (2012) Remarks on cannibalism and pattern formation in spatially extended prey–predator systems. Nonlinear Dyn 67(4):2543–2548MathSciNetzbMATHGoogle Scholar
  63. 63.
    Basheer A, Parshad RD, Quansah E, Shengbin Y, Upadhyay RK (2018) Exploring the dynamics of a Holling–Tanner model with cannibalism in both predator and prey population. Int J Biomath 11(01):1850010MathSciNetzbMATHGoogle Scholar
  64. 64.
    Tripathi JP, Abbas S, Thakur M (2015) A density dependent delayed predator–prey model with Beddington–DeAngelis type function response incorporating a prey refuge. Commun Nonlinear Sci Numer Simul 22(1–3):427–450MathSciNetzbMATHGoogle Scholar
  65. 65.
    Tripathi JP, Abbas S, Thakur M (2015) Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge. Nonlinear Dyn 80(1–2):177–196MathSciNetzbMATHGoogle Scholar
  66. 66.
    Tripathi JP, Tyagi S, Abbas S (2016) Global analysis of a delayed density dependent predator–prey model with Crowley–Martin functional response. Commun Nonlinear Sci Numer Simul 30(1–3):45–69MathSciNetGoogle Scholar
  67. 67.
    Abbas S, Tripathi JP, Neha AA (2017) Dynamical analysis of a model of social behavior: criminal versus non-criminal population. Chaos Solitons Fractals 98:121–129MathSciNetzbMATHGoogle Scholar
  68. 68.
    Křivan V (1996) Optimal foraging and predator–prey dynamics. Theor Popul Biol 49(3):265–290zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Aladeen Basheer
    • 1
  • Emmanuel Quansah
    • 2
  • Rana D. Parshad
    • 3
    Email author
  1. 1.Mathematics DepartmentUniversity of GeorgiaAthensUSA
  2. 2.Factory Mutual InsuranceJohnstonUSA
  3. 3.Department of MathematicsIowa State UniversityAmesUSA

Personalised recommendations