Complex coexistence induced by delays in a diffusive intraguild predation system with stage structure and spatial memory

Shuai Li; Bin Fang; Xinyu Song; Chengdai Huang; Shuai Li; Bin Fang; Xinyu Song; Chengdai Huang

doi:10.3934/math.2026266

AIMS Mathematics

2026, Volume 11, Issue 3: 6437-6463. doi: 10.3934/math.2026266

Previous Article Next Article

Research article Topical Sections

Complex coexistence induced by delays in a diffusive intraguild predation system with stage structure and spatial memory

1.
College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
2.
Henan Provincial Center for Applied Mathematics, Xinyang Normal University, Xinyang 464000, China

Received: 15 January 2026 Revised: 27 February 2026 Accepted: 11 March 2026 Published: 13 March 2026
MSC : 37L10, 92B05, 92D25

We study Hopf bifurcations induced by spatial memory and stage structure in a diffusive intraguild predation system, aiming to reveal their effects on the spatiotemporal dynamics of three interacting species. First, by applying the geometric approach, we obtain the stability criteria and Hopf bifurcation conditions through an analysis of the linearized system with delay-dependent coefficients. Next, we develop a computational algorithm for the normal form of Hopf bifurcation induced by maturation delay in a system with three state variables and diffusion terms containing a delay. Finally, we combine numerical simulations to demonstrate that variations in periods of maturation and spatial memory can generate rich and complex coexistence dynamics, including spatially homogeneous periodic solutions, spatially heterogeneous periodic solutions with mode$ -5 $, and multiple stability switches.
- delays,
- diffusive predator-prey system,
- stability,
- Hopf bifurcation,
- normal form
Citation: Shuai Li, Bin Fang, Xinyu Song, Chengdai Huang. Complex coexistence induced by delays in a diffusive intraguild predation system with stage structure and spatial memory[J]. AIMS Mathematics, 2026, 11(3): 6437-6463. doi: 10.3934/math.2026266

Related Papers:

Abstract

We study Hopf bifurcations induced by spatial memory and stage structure in a diffusive intraguild predation system, aiming to reveal their effects on the spatiotemporal dynamics of three interacting species. First, by applying the geometric approach, we obtain the stability criteria and Hopf bifurcation conditions through an analysis of the linearized system with delay-dependent coefficients. Next, we develop a computational algorithm for the normal form of Hopf bifurcation induced by maturation delay in a system with three state variables and diffusion terms containing a delay. Finally, we combine numerical simulations to demonstrate that variations in periods of maturation and spatial memory can generate rich and complex coexistence dynamics, including spatially homogeneous periodic solutions, spatially heterogeneous periodic solutions with mode$ -5 $, and multiple stability switches.

References

[1]	S. Gao, H. Wang, S. Yuan, Theory of stoichiometric intraguild predation: algae, ciliate, and daphnia, Bull. Math. Biol., 86 (2024), 79. https://doi.org/10.1007/s11538-024-01306-z doi: 10.1007/s11538-024-01306-z
[2]	C. A. M. Hickerson, C. D. Anthony, B. M. Walton, Edge effects and intraguild predation in native and introduced centipedes: evidence from the field and from laboratory microcosms, Oecologia, 146 (2005), 110–119. https://doi.org/10.1007/s00442-005-0197-y doi: 10.1007/s00442-005-0197-y
[3]	P. Morin, Productivity, intraguild predation, and population dynamics in experimental food webs, Ecology, 80 (1999), 752–760. https://doi.org/10.1890/0012-9658(1999)080[0752:PIPAPD]2.0.CO;2 doi: 10.1890/0012-9658(1999)080[0752:PIPAPD]2.0.CO;2
[4]	Y. Kang, L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, J. Math. Biol., 67 (2013), 1227–1259. https://doi.org/10.1007/s00285-012-0584-z doi: 10.1007/s00285-012-0584-z
[5]	G. A. Polis, R. D. Holt, Intraguild predation: the dynamics of complex trophic interactions, Trends Ecol. Evol., 7 (1992), 151–154. https://doi.org/10.1016/0169-5347(92)90208-S doi: 10.1016/0169-5347(92)90208-S
[6]	R. D. Holt, G. A. Polis, A theoretical framework for intraguild predation, Amer. Nat., 149 (1997), 745–764. https://doi.org/10.1086/286018 doi: 10.1086/286018
[7]	U. Kumar, P. S. Mandal, Impact of additive Allee effect on the dynamics of an intraguild predation model with specialist predator, Int. J. Bifurcat. Chaos, 30 (2020), 2050239. https://doi.org/10.1142/S0218127420502399 doi: 10.1142/S0218127420502399
[8]	J. Ji, L. Wang, Competitive exclusion and coexistence in an intraguild predation model with Beddington–DeAngelis functional response, Commun. Nonlinear Sci. Numer. Simul., 107 (2022), 106192. https://doi.org/10.1016/j.cnsns.2021.106192 doi: 10.1016/j.cnsns.2021.106192
[9]	Y. Kuang, Delay differential equations: with applications in population dynamics, Academic Press, 1993.
[10]	S. A. Gourley, Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188–200, https://doi.org/10.1007/s00285-004-0278-2 doi: 10.1007/s00285-004-0278-2
[11]	H. Shu, X. Hu, L. Wang, J. Watmough, Delay induced stability switch, multitype bistability and chaos in an intraguild predation model, J. Math. Biol., 71 (2015), 1269–1298. https://doi.org/10.1007/s00285-015-0857-4 doi: 10.1007/s00285-015-0857-4
[12]	M. Yamaguchi, Y. Takeuchi, W. Ma, Dynamical properties of a stage structured three-species model with intra-guild predation, J. Comput. Appl. Math., 201 (2007), 327–338. https://doi.org/10.1016/j.cam.2005.12.033 doi: 10.1016/j.cam.2005.12.033
[13]	S. Li, Y. Yang, X. Song, The joint impacts of maturation delay and digestion delay on the dynamics of a predator–prey model, J. Xinyang Norm. Univ. (Nat. Sci. Ed.), 38 (2025), 421–427. https://doi.org/10.3969/j.issn.2097-583X.2025.04.007 doi: 10.3969/j.issn.2097-583X.2025.04.007
[14]	W. Xu, H. Shu, L. Wang, G. Fan, Bifurcation-induced species coexistence in a stage-structured intraguild predation model, SIAM J. Appl. Math., 85 (2025), 662–686. https://doi.org/10.1137/24M1640392 doi: 10.1137/24M1640392
[15]	R. S. Cantrell, C. Cosner, Spatial ecology via reaction–Diffusion equations, John Wiley & Sons, 2004. https://doi.org/10.1002/0470871296
[16]	J. R. Potts, M. A. Lewis, Spatial memory and taxis-driven pattern formation in model ecosystems, Bull. Math. Biol., 81 (2019), 2725–2747. https://doi.org/10.1007/s11538-019-00626-9 doi: 10.1007/s11538-019-00626-9
[17]	B. Zhang, K. Y. Lam, W. M. Ni, R. Signorelli, K. M. Collins, Z. Fu, et al., Directed movement changes coexistence outcomes in heterogeneous environments, Ecol. Lett., 25 (2022), 366–377. https://doi.org/10.1111/ele.13925 doi: 10.1111/ele.13925
[18]	Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differ. Equ., 223 (2006), 400–426. https://doi.org/10.1016/j.jde.2005.05.010 doi: 10.1016/j.jde.2005.05.010
[19]	G. Q. Sun, J. Zhang, L. P. Song, Z. Jin, B. L. Li, Pattern formation of a spatial predator–prey system, Appl. Math. Comput., 218 (2012), 11151–11162. https://doi.org/10.1016/j.amc.2012.04.071 doi: 10.1016/j.amc.2012.04.071
[20]	W. Xu, H. Shu, Z. Tang, H. Wang, Complex dynamics in a general diffusive predator–prey model with predator maturation delay, J. Dyn. Diff. Equat., 36 (2024), 1879–1904. https://doi.org/10.1007/s10884-022-10176-9 doi: 10.1007/s10884-022-10176-9
[21]	D. Ryan, R. S. Cantrell, Avoidance behavior in intraguild predation communities: a cross-diffusion model, Discrete Contin. Dyn. Syst., 35 (2015), 1641–1663. https://doi.org/10.3934/dcds.2015.35.1641 doi: 10.3934/dcds.2015.35.1641
[22]	F. Farivar, G. Gambino, V. Giunta, M. C. Lombardo, M. Sammartino, Intraguild predation communities with anti–predator behavior, SIAM J. Appl. Dyn. Syst., 24 (2025), 1110–1149. https://doi.org/10.1137/24M1632747 doi: 10.1137/24M1632747
[23]	B. Abrahms, E. L. Hazen, E. O. Aikens, M. S. Savoca, J. A. Goldbogen, S. J. Bograd, et al., Memory and resource tracking drive blue whale migrations, Proc. Natl. Acad. Sci. USA, 116 (2019), 5582–5587. https://doi.org/10.1073/pnas.1819031116 doi: 10.1073/pnas.1819031116
[24]	H. Wang, Y. Salmaniw, Open problems in PDE models for knowledge-based animal movement via nonlocal perception and cognitive mapping, J. Math. Biol., 86 (2023), 71. https://doi.org/10.1007/s00285-023-01905-9 doi: 10.1007/s00285-023-01905-9
[25]	S. Li, Q. Wei, X. Song, L. Dai, C. Huang, Double Hopf bifurcation generated by dual memory delays in a diffusive two-species model, J. Appl. Math. Comput., 72 (2026), 101. https://doi.org/10.1007/s12190-025-02754-z doi: 10.1007/s12190-025-02754-z
[26]	J. A. Merkle, H. Sawyer, K. L. Monteith, S. P. Dwinnell, G. L. Fralick, M. J. Kauffman, Spatial memory shapes migration and its benefits: evidence from a large herbivore, Ecol. Lett., 22 (2019), 1797–1805. https://doi.org/10.1111/ele.13362 doi: 10.1111/ele.13362
[27]	J. Shi, C. Wang, H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, 32 (2019), 3188. https://doi.org/10.1088/1361-6544/ab1f2f doi: 10.1088/1361-6544/ab1f2f
[28]	C. Wang, S. Yuan, H. Wang, Spatiotemporal patterns of a diffusive prey-predator model with spatial memory and pregnancy period in an intimidatory environment, J. Math. Biol., 84 (2022), 12. https://doi.org/10.1007/s00285-022-01716-4 doi: 10.1007/s00285-022-01716-4
[29]	M. Banerjee, S. Petrovskii, Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system, Theor. Ecol., 4 (2011), 37–53. https://doi.org/10.1007/s12080-010-0073-1 doi: 10.1007/s12080-010-0073-1
[30]	J. Xu, T. Zou, Dynamics of a predator-prey model with the Allee effect for the predator induced by weighted harvesting strategy, AIMS Math., 11 (2026), 578–593. https://doi.org/10.3934/math.2026025 doi: 10.3934/math.2026025
[31]	Y. Song, Y. Peng, T. Zhang, Dynamics of a predator-prey model with the Allee effect for the predator induced by weighted harvesting strategy, J. Differ. Equ., 300 (2021), 597–624. https://doi.org/10.1016/j.jde.2021.08.010 doi: 10.1016/j.jde.2021.08.010
[32]	M. Liu, H. Wang, W. Jiang, Double-Hopf bifurcation and bistable asynchronous periodic orbits for the memory-based diffusion system, SIAM J. Appl. Dyn. Syst., 23 (2024), 2732–2768. https://doi.org/10.1137/23M1627493 doi: 10.1137/23M1627493
[33]	Q. An, E. Beretta, Y. Kuang, C. Wang, H. Wang, Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters, J. Differ. Equ., 266 (2019), 7073–7100. https://doi.org/10.1016/j.jde.2018.11.025 doi: 10.1016/j.jde.2018.11.025
[34]	Y. Ding, P. Yu, Dynamics of a pine wilt disease control model with nonlocal competition and memory diffusion, Math. Biosci., 389 (2025), 109524. https://doi.org/10.1016/j.mbs.2025.109524 doi: 10.1016/j.mbs.2025.109524
[35]	J. Shi, C. Wang, H. Wang, Spatial movement with diffusion and memory-based self-diffusion and cross-diffusion, J. Differ. Equ., 305 (2021), 242–269. https://doi.org/10.1016/j.jde.2021.10.021 doi: 10.1016/j.jde.2021.10.021
[36]	T. Zheng, L. Zhang, Y. Luo, X. Zhou, H. L. Li, Z. Teng, Stability and Hopf bifurcation of a stage-structured cannibalism model with two delays, Int. J. Bifurcat. Chaos, 31 (2021), 2150242. https://doi.org/10.1142/S0218127421502424 doi: 10.1142/S0218127421502424
[37]	T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217–2238. https://doi.org/10.1090/S0002-9947-00-02280-7 doi: 10.1090/S0002-9947-00-02280-7
[38]	S. Li, S. Yuan, Z. Jin, H. Wang, Double Hopf bifurcation induced by spatial memory in a diffusive predator–prey model with Allee effect and maturation delay of predator, Commun. Nonlinear Sci. Numer. Simul., 132 (2024), 107936. https://doi.org/10.1016/j.cnsns.2024.107936 doi: 10.1016/j.cnsns.2024.107936
[39]	Y. Lv, Global boundedness and spatially inhomogeneous Hopf bifurcation in a delayed predator–prey model with dual taxis, Chaos Soliton. Fract., 201 (2025), 117369. https://doi.org/10.1016/j.chaos.2025.117369 doi: 10.1016/j.chaos.2025.117369
[40]	X. Chen, M. Gao, Y. Jiang, D. Jiang, Dynamic properties of a prey–predator food chain chemostat model with Ornstein–Uhlenbeck process, Math. Methods Appl. Sci., 48 (2025), 9253–9271. https://doi.org/10.1002/mma.10797 doi: 10.1002/mma.10797
[41]	M. Gao, D. Jiang, J. Ding, Analysis of a chemostat model with species death rate and Ornstein–Uhlenbeck process, J. Math. Phys., 66 (2025), 062702. https://doi.org/10.1063/5.0214134 doi: 10.1063/5.0214134
[42]	M. Gao, X. Chen, D. Jiang, Exploring the impact of distributed delay and environmental fluctuation on food chain stability in a chemostat model, Qual. Theory Dyn. Syst., 24 (2025), 149. https://doi.org/10.1007/s12346-025-01303-0 doi: 10.1007/s12346-025-01303-0

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)