Skip to main content
SpringerLink
Log in

Spatiotemporal dynamics induced by intraguild predator diffusion in an intraguild predation model

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

An intraguild predation model with intraguild predator diffusion is proposed and studied in this work. It is shown that the local system can have four boundary equilibria and at most two interior equilibria. The interior equilibria may exist even when the system is not uniformly persistent. When only intraguild predator diffusion is incorporated into our three-species model, the resulting model is a partially degenerate reaction-diffusion system. For this partially degenerate system, we show that the solution semiflow is bounded dissipative and the positive orbits of bounded sets are bounded. We also demonstrate that intraguild predator diffusion can lead to the occurrence of spatially nonhomogeneous oscillations and spatiotemporal chaos. Further, we show that intraguild predator diffusion can induce transitions between spatially homogeneous oscillations, spatially nonhomogeneous oscillations and chaos.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Abrams PA, Fung SR (2010) Prey persistence and abundance in systems with intraguild predation and type-2 functional responses. J Theor Biol 264:1033–1042

    Article  MathSciNet  MATH  Google Scholar 

  • Amarasekare P (2007) Spatial dynamics of communities with intraguild predation: the role of dispersal strategies. Am Nat 170:819–831

    Article  Google Scholar 

  • Beddington JR (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J Animal Ecol 44:331–340

    Article  Google Scholar 

  • Conley C (1978) Isolated invariant sets and the Morse index, CBMS regional conference series in mathematics, American Mathematical Society, Providence, R.I. 38

  • DeAngelis DL, Goldstein RA, Neill R (1975) A model for trophic interaction. Ecology 56:881–892

    Article  Google Scholar 

  • Drolet D, Barbeau MA, Coffin MRS, Hamilton DJ (2009) Effect of the snail Ilyanassaobsoleta (Say) on dynamics of the amphipod Corophium volutator (Pallas) on an intertidal mudflat. J Exp Mar Biol Ecol 368:189–195

    Article  Google Scholar 

  • Du Z, Peng R (2016) A priori \(L^{\infty }\) estimates for solutions of a class of reaction-diffusion system. J Math Biol 72:1429–1439

    Article  MathSciNet  MATH  Google Scholar 

  • Fedriani JM, Fuller TK, Sauvajot RM, York EC (2000) Competition and intraguild predation among three sympatric carnivores. Oecologia 125:258–270

    Article  Google Scholar 

  • Freedman HI, Ruan S, Tang M (1994) Uniform persistence and flows near a closed positively invariant set. J Dyn Differ Equ 6:583–600

    Article  MathSciNet  MATH  Google Scholar 

  • Freedman HI, Waltman P (1984) Persistence in models of three interacting predator-prey populations. Math Biosci 68:213–231

    Article  MathSciNet  MATH  Google Scholar 

  • Guckenheimer T, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Sci. 42. Springer-Verlag, New York

  • Hale JK (1988) Asymptotic behavior of dissipative systems. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Han R, Dai B (2017) Spatiotemporal dynamics and spatial pattern in a diffusive intraguild predation model with delay effect. Appl Math Comput 312:177–201

    MathSciNet  MATH  Google Scholar 

  • Han R, Dai B, Chen Y (2019) Pattern formation in a diffusive intraguild predation model with nonlocal interaction effects. AIP Advances 9:035046

    Article  Google Scholar 

  • Hastings A, Abbott KC, Cuddington K et al. (2018) Transient phenomena in ecology, Science, 361: eaat6412

  • Hattaf K, Yousfi N (2015) A generalized HBV model with diffusion and two delays. Comput Math Appl 69:31–40

    Article  MathSciNet  MATH  Google Scholar 

  • Henrot A (2006) Extremum problems for eigenvalues of elliptic operators, Front. Math., Birkhäuser Verlag, Basel

  • Holling CS (1959) The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can Ent 91:293–320

    Article  Google Scholar 

  • Holling CS (1959) Some characteristics of simple types of predation and parasitism. Can Ent 91:385–398

    Article  Google Scholar 

  • Holling CS (1966) The functional response of invertebrate predators to prey density. Mem Ent Soc Can 98:5–86

    Article  Google Scholar 

  • Holt RD, Polis GA (1997) A theoretical framework for intraguild predation. Am Nat 149:745–764

    Article  Google Scholar 

  • Hsu S, Ruan S, Yang T (2015) Analysis of three species Lotka-Volterra food web models with omnivory. J Math Anal Appl 426:659–687

    Article  MathSciNet  MATH  Google Scholar 

  • Ji J, Wang L (2022) Competitive exclusion and coexistence in an intraguild predation model with Beddington-DeAngelis functional response. Commun Nonlinear Sci Numer Simul 107:106192

    Article  MathSciNet  MATH  Google Scholar 

  • Kang Y, Wedekin L (2012) Dynamics of a intraguild predation model with generalist or specialist predator. J Math Biol 67:1227–1259

    Article  MathSciNet  MATH  Google Scholar 

  • Lester PJ, Harmsen R (2002) Functional and numerical responses do not always indicate the most effective predator for biological control: an analysis of two predators in a two-prey system. J Appl Ecol 39:455–468

    Article  Google Scholar 

  • Lin J, Yang T (2018) Traveling wave solutions for a diffusive three-species intraguild predation model. Int J Biomath 11:1850022

    Article  MathSciNet  MATH  Google Scholar 

  • Liu WM (1994) Criterion of Hopf bifurcations without using eigenvalues. J Math Anal Appl 182:250–256

    Article  MathSciNet  MATH  Google Scholar 

  • Ma Z, Shu H (2020) Viral infection dynamics in a spatial heterogeneous environment with cell-free and cell-to-cell transmissions. Math Biosci Eng 17:2569–2591

    Article  MathSciNet  MATH  Google Scholar 

  • Martin RH, Smith HL (1990) Abstract functional differential equations and reaction-diffusion systems. Trans Amer Math Soc 321:1–44

    MathSciNet  MATH  Google Scholar 

  • Morozov A, Abbott K, Cuddington K, Francis T et al (2020) Long transients in ecology: theory and applications. Phys Life Rev 32:1–40

    Article  Google Scholar 

  • Polis GA, Holt RD (1992) Intraguild predation: the dynamics of complex trophic interactions. Trends Ecol Evol 7:151–154

    Article  Google Scholar 

  • Polis GA, Myers CA, Holt RD (1989) The ecology and evolution of intraguild predation: potential competitors that eat each other. Annu Rev Ecol Syst 20:297–230

    Article  Google Scholar 

  • Protter MH, Weinberger HF (2012) Maximum principles in differential equations, Springer

  • Sen D, Ghorai S, Banerjee M (2018) Complex dynamics of a three species prey-predator model with intraguild predation. Ecol Complex 34:9–22

    Article  Google Scholar 

  • Shu H, Hu X, Wang L, Watmough J (2015) Delay induced stability switch, multi-type bistability and chaos in an intraguild predation model. J Math Biol 71:1269–1298

    Article  MathSciNet  MATH  Google Scholar 

  • Shu H, Ma Z, Wang X, Wang L (2020) Viral diffusion and cell-to-cell transmission: mathematical analysis and simulation study. J Math Pures Appl 137:290–313

    Article  MathSciNet  MATH  Google Scholar 

  • Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Smith TM, Smith RL (2014) Elements of ecology, 9th Edition. Pearson

  • Solomon ME (1949) The natural control of animal populations. J Anim Ecol 19:1–35

    Article  Google Scholar 

  • Tanabe K, Namba T (2005) Omnivory creates chaos in simple food web models. Ecology 86:3411–3414

    Article  Google Scholar 

  • Velazquez I, Kaplan D, Velasco-Hernandez JX, Navarrete SA (2005) Multistability in an open recruitment food web model. J Appl Math Comput 163:275–294

    Article  MATH  Google Scholar 

  • Verdy A, Amarasekare P (2010) Alternative stable states in communities with intraguild predation. J Theor Biol 262:116–128

    Article  MathSciNet  MATH  Google Scholar 

  • Wang F, Huang Y, Zou X (2014) Global dynamics of a PDE in-host viral model. Appl Anal 93:2312–2329

    Article  MathSciNet  MATH  Google Scholar 

  • Wang K, Wang W (2007) Propagation of HBV with spatial dependence. Math Biosci 21:78–95

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Watmough J, Yu F (2021) Dynamics of a plant-herbivore system with chemically mediated numerical response. Math Appl Sci Eng 2:72–148

    Google Scholar 

  • Wang X, Zhao X-Q (2015) Pulsating waves of a partially degenerate reaction-diffusion system in a periodic habitat. J Differ Equ 259:7238–7259

    Article  MathSciNet  MATH  Google Scholar 

  • Wei H (2019) A mathematical model of intraguild predation with prey switching. Math Comput Simul 165:107–118

    Article  MathSciNet  MATH  Google Scholar 

  • Wu Y, Zou X (2018) Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates. J Differ Equ 264:4989–5024

    Article  MathSciNet  MATH  Google Scholar 

  • Yamaguchi M, Takeuchi Y, Ma W (2007) Dynamical properties of a stage structured three-species model with intra-guild predation. J Comput Appl Math 201:327–338

    Article  MathSciNet  MATH  Google Scholar 

  • Yang T, Zhang W, Cheng K (2016) Global dynamics of three species omnivory models with Lotka-Volterra interaction. Discrete Contin Dyn Syst Ser B 21:2867–2881

    Article  MathSciNet  MATH  Google Scholar 

  • Yu F, Wang L, Watmough J (2016) Transient spatio-temporal dynamics of a diffusive plant-herbivore system with Neumann boundary conditions. J Biol Dyn 10:477–500

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang D, Dai B (2019) A free boundary problem for the diffusive intraguild predation model with intraspecific competition. J Math Anal Appl 474:381–412

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang D, Dai B (2019) Spreading and vanishing in a diffusive intraguild predation model with intraspecific competition and free boundary. Math Methods Appl Sci 42:6917–6943

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang D, Dai B (2021) The diffusive intraguild predation model with intraspecific competition and double free boundaries. Appl Anal 100:3322–3349

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

GL’s work was partially supported by the National Natural Science Foundation of China (No. 12001127) and the Research Project of Guangzhou University (No. RP2021035). AM’s work was partially supported by the National Natural Science Foundation of China (No. 12101547), the Natural Science Foundation of Shanxi Province (No. 201901D111295) and Shanxi Scholarship Council of China (No: 2021-149). JJ acknowledges the support from Dr. William S. Lewis Doctoral Fellowship at the University of New Brunswick. LW’s work was partially supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2020-04143).

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

    Juping Ji & Lin Wang

  2. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Genghong Lin

  3. Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, China

    Genghong Lin

  4. School of Mathematics and Information Technology, Yuncheng University, Yuncheng, 044000, China

    Ali Mai

Authors
  1. Juping Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Genghong Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ali Mai
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. The first draft of the manuscript was written by Juping Ji and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Juping Ji.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ji, J., Lin, G., Wang, L. et al. Spatiotemporal dynamics induced by intraguild predator diffusion in an intraguild predation model. J. Math. Biol. 85, 1 (2022). https://doi.org/10.1007/s00285-022-01772-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00285-022-01772-w

Keywords

Mathematics Subject Classification

Search

Navigation