pith. sign in

arxiv: 2604.18129 · v2 · submitted 2026-04-20 · 🧮 math.AP

Global dynamics and diffusion-driven pattern formation in a predator-prey system with two chemicals

Gnanasekaran Shanmugasundaram , Jitraj Saha , Oluwole Daniel Makinde , Joydev Chattopadhyay This is my paper

Pith reviewed 2026-05-10 04:12 UTC · model grok-4.3

classification 🧮 math.AP
keywords predator-prey modelcross diffusionTuring patternsLyapunov stabilityglobal existencereaction diffusion systemchemical mediationdiffusion driven instability
0
0 comments X

The pith

Global existence and stability hold for a predator-prey system with two chemicals and cross-diffusion, with numerical evidence of diffusion-driven patterns.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that a system of four coupled reaction-diffusion equations for prey, predator, and two chemicals has global classical solutions when parameters meet certain conditions. A Lyapunov functional is built to prove that the uniform steady state is asymptotically stable. Numerical methods are used to find Turing bifurcation points, and simulations demonstrate how changes in the predation rate affect whether spatial patterns appear. A reader would care because these results indicate that chemical interactions combined with diffusion can determine whether populations stay mixed or organize into spatial patterns over long times.

Core claim

Under appropriate conditions on the model parameters, the global existence of classical solutions is established for the predator-prey cross-diffusion system coupled with two chemical substances in a bounded domain. By constructing a suitable Lyapunov functional, the asymptotic stability of the spatially homogeneous steady state is proved. The emergence of spatial patterns induced by diffusion-driven instability is investigated numerically because analytical derivation is infeasible due to system complexity, with simulations producing bifurcation diagrams that show the system's response to predation rate variations.

What carries the argument

The cross-diffusion terms in the four-equation system that couple the population densities with the chemical concentrations, together with the Lyapunov functional constructed to control the deviation from the steady state.

Load-bearing premise

The model parameters are restricted to values that prevent solutions from becoming unbounded in finite time, though the precise bounds are not provided explicitly.

What would settle it

A counterexample consisting of specific parameter values and initial conditions where the solution develops a singularity in finite time, or where the computed Lyapunov derivative does not remain non-positive.

Figures

Figures reproduced from arXiv: 2604.18129 by Gnanasekaran Shanmugasundaram, Jitraj Saha, Joydev Chattopadhyay, Oluwole Daniel Makinde.

Figure 1
Figure 1. Figure 1: Parameters µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, a1 = a2 = b1 = b2 = 0.5, η1 = 10, η2 = 2. Turing patterns arise when diffusion destabilizes a spatially homogeneous steady state, lead￾ing to the spontaneous formation of stationary structures such as spots, stripes, or patches. In reaction–diffusion systems, this diffusion-driven instability amplifies specific spatial modes, producing ordered patterns without ext… view at source ↗
Figure 2
Figure 2. Figure 2: Phase portrait of the species and chemicals. Parameter values µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, a1 = a2 = 1, b1 = b2 = 2, η1 = 10, η2 = 2. The initial condition is u10 = 1, u20 = 2, v10 = v20 = 0. The above figure presents the phase portraits of the predator–prey system and their associated chemical signals for the given parameter set, where the trajectories represent the temporal evolution of the interactin… view at source ↗
Figure 3
Figure 3. Figure 3: LAS of the coexistence state of system (2.1) for the parameter values d11 = 0.1, d12 = 0, d3 = 3, d21 = 1, d22 = 0, d4 = 2, µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, a1 = a2 = b1 = b2 = 0.5, η1 = 10, η2 = 2 and t = 1000. Therefore, the coexistence steady state (u ∗ 1 , u∗ 2 , v∗ 1 , v∗ 2 ) is locally asymptotically stable. More￾over, both the corresponding ODE system and the reaction–diffusion system incorporating s… view at source ↗
Figure 4
Figure 4. Figure 4: Cross-diffusion driven instability of the system (2.1) for the parameter values d11 = 0.1, d12 = 1, d3 = 3, d21 = 1, d22 = 2, d4 = 2, µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, a1 = a2 = b1 = b2 = 0.5, η1 = 10, η2 = 2 and t = 1000. We now introduce the finite difference method to conduct numerical simulations and inves￾tigate the temporal and spatial dynamics of the predator–prey system (2.1). We consider the square … view at source ↗
Figure 5
Figure 5. Figure 5: (a) Dispersion relation and (b) h(k 2 ) with respect to k 2 , where d11 = 0.1, d12 = 1, d3 = 3, d21 = 1, d22 = 2, d4 = 2, µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, a1 = a2 = b1 = b2 = 0.5, η1 = 10 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spatial distributions of the species Predator u1 (first row) and Prey u2 (second row), where d11 = 0.1, d12 = 1, d3 = 3, d21 = 1, d22 = 2, d4 = 2, µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, a1 = a2 = b1 = b2 = 0.5, η1 = 10. 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 1.05000 1.10000 1.15000 1.20000 1.25000 1.30000 1.35000 1.40000 1.45000 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 1.05000 1.1000… view at source ↗
Figure 7
Figure 7. Figure 7: Spatial distributions of the species Predator u1 (first row) and Prey u2 (second row), where d11 = 0.1, d12 = 1, d21 = 1, d22 = 2, d3 = 3, d4 = 2, µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, a1 = a2 = b1 = b2 = 0.5, η1 = 10 [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Spatial distributions of the species Predator u1 (first row) and Prey u2 (second row), where d11 = 3.25, d12 = 3, d3 = 0.75, d21 = 5, d22 = 3, d4 = 0.75, a1 = a2 = b1 = b2 = 0.5, µ1 = 2, µ2 = 1, λ1 = 2, λ2 = 1. 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 5.00000 10.00000 15.00000 20.00000 25.00000 30.00000 35.00000 40.00000 45.00000 50.00000 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200… view at source ↗
Figure 9
Figure 9. Figure 9: Spatial distributions of the species Predator u1 (first row) and Prey u2 (second row), where d11 = 3.25, d12 = 3, d3 = 0.75, d21 = 5, d22 = 3, d4 = 0.75, a1 = a2 = b1 = b2 = 0.5, µ1 = 0.5, µ2 = 0.1, λ1 = 0.5, λ2 = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Spatial distributions of the species Predator u1 (first row) and Prey u2 (second row), where d11 = 1, d12 = 1, d3 = 1, d21 = 5, d22 = 5, d4 = 1.5, a1 = a2 = b1 = b2 = 0.5, µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, η1 = 10. 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 1.20000 1.40000 1.60000 1.80000 2.00000 2.20000 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 1.20000 1.40000 1.60000 1.80000 2.000… view at source ↗
Figure 11
Figure 11. Figure 11: Spatial distributions of the species Predator u1 (first row) and Prey u2 (second row), where (a, c) d11 = 0.1, d12 = 1, d21 = 0.1, d22 = 1, d3 = 3, d4 = 3, µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, a1 = a2 = b1 = b2 = 5, η1 = 5, η2 = 1.5. (b, d) d11 = 0.1, d12 = 1, d21 = 1, d22 = 2, d3 = 3, d4 = 2, µ1 = 2, µ2 = 3, λ1 = 2, λ2 = 1, a1 = a2 = b1 = b2 = 5, η1 = 5, η2 = 1.5 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
read the original abstract

This work analyzes a predator-prey cross-diffusion system coupled with two chemical substances under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R^n (n >= 2) with smooth boundary dOmega. Under appropriate conditions on the model parameters, the global existence of classical solutions is established. Furthermore, by constructing a suitable Lyapunov functional, the asymptotic stability of the spatially homogeneous steady state is proved. The emergence of spatial patterns induced by diffusion-driven instability is also investigated. Owing to the complexity of the resulting four-equation system, the criteria for Turing bifurcation are derived numerically rather than analytically. Numerical simulations are performed to generate Turing bifurcation diagrams, illustrating the dynamical responses of the system to variations in the predation rate. These results provide new insights into the role of predation intensity in the formation of spatial patterns in predator-prey systems mediated by two chemical substances.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper analyzes a four-equation predator-prey cross-diffusion system with two chemicals under homogeneous Neumann boundary conditions in a bounded domain. It claims global existence of classical solutions under appropriate parameter conditions, proves asymptotic stability of the spatially homogeneous steady state via a constructed Lyapunov functional, and investigates diffusion-driven instability by numerically computing Turing bifurcation criteria (due to system complexity), supported by simulations of dynamical responses to predation rate variations.

Significance. If the claims hold with the required details, the work extends standard Lyapunov techniques and numerical bifurcation analysis to a coupled four-component cross-diffusion model, offering insights into how predation intensity influences spatial pattern formation in chemically mediated ecological systems. The combination of global dynamics and Turing instability exploration in this setting could inform further studies on multi-species reaction-diffusion systems.

major comments (2)
  1. [Abstract and global existence section] Abstract and global existence theorem: the claim of global existence of classical solutions 'under appropriate conditions on the model parameters' is not accompanied by explicit inequalities, ranges, or assumptions on the diffusion coefficients, reaction rates, or cross-diffusion terms. This vagueness is load-bearing, as it prevents verification of the a priori estimates used to establish the result.
  2. [Section on diffusion-driven instability and numerical simulations] Turing bifurcation analysis: the criteria for diffusion-driven instability are derived numerically from the degree-4 characteristic polynomial of the linearized four-equation system, but no mesh-refinement studies, root-finding tolerance bounds, or comparisons to analytically tractable reduced subsystems are reported. This is load-bearing for the pattern formation claim, as it leaves open whether the bifurcation diagrams reliably identify instability boundaries or reflect discretization artifacts.
minor comments (1)
  1. [Model formulation] The model equations in the introduction could benefit from explicit notation for all cross-diffusion and chemical interaction terms to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive major comments. We address each point below and indicate the revisions planned for the next version.

read point-by-point responses

  1. Referee: [Abstract and global existence section] Abstract and global existence theorem: the claim of global existence of classical solutions 'under appropriate conditions on the model parameters' is not accompanied by explicit inequalities, ranges, or assumptions on the diffusion coefficients, reaction rates, or cross-diffusion terms. This vagueness is load-bearing, as it prevents verification of the a priori estimates used to establish the result.

    Authors: The global existence result (Theorem 3.1) is stated under explicit assumptions: all self-diffusion coefficients are positive, cross-diffusion coefficients are non-negative, and reaction rates satisfy standard positivity and quadratic growth bounds that permit application of the maximum principle followed by energy estimates. The abstract employs the phrase 'appropriate conditions' purely as a summary. To improve readability and address the concern directly, we will revise the abstract to read 'under the assumptions that all diffusion coefficients are positive and the cross-diffusion coefficients are bounded' and add a short clarifying sentence in the theorem statement listing the key parameter restrictions. revision: yes

  2. Referee: [Section on diffusion-driven instability and numerical simulations] Turing bifurcation analysis: the criteria for diffusion-driven instability are derived numerically from the degree-4 characteristic polynomial of the linearized four-equation system, but no mesh-refinement studies, root-finding tolerance bounds, or comparisons to analytically tractable reduced subsystems are reported. This is load-bearing for the pattern formation claim, as it leaves open whether the bifurcation diagrams reliably identify instability boundaries or reflect discretization artifacts.

    Authors: We agree that additional numerical validation strengthens the Turing analysis. The quartic characteristic polynomial is solved using a standard eigenvalue routine at machine precision. In the revised manuscript we will document the root-finding tolerance (10^{-12}), include a mesh-refinement study comparing bifurcation curves computed on 100-by-100 and 200-by-200 parameter grids to confirm convergence of the instability region, and add a brief comparison with the reduced two-chemical subsystem obtained by setting one chemical to its quasi-steady state, showing that the qualitative location of the Turing boundary is preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper establishes global existence of classical solutions under appropriate parameter conditions via standard a priori estimates for cross-diffusion PDE systems. Asymptotic stability of the homogeneous steady state is shown by constructing an independent Lyapunov functional whose form does not presuppose the pattern-formation results. Turing bifurcation criteria are obtained numerically because the four-equation linearization is analytically intractable; these are presented explicitly as simulation outputs and bifurcation diagrams rather than as fitted predictions or relabeled inputs. No load-bearing self-citations, self-definitional steps, or ansatz smuggling appear in the provided derivation chain. The numerical component is a computational investigation, not a statistical fit renamed as a prediction, so the overall argument does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No specific free parameters or invented entities are detailed in the abstract. The work relies on standard assumptions from PDE theory for reaction-diffusion systems.

axioms (2)
  • standard math Existence of classical solutions for the parabolic system under suitable parameter conditions
    Stated for global existence in the abstract.
  • domain assumption Homogeneous Neumann boundary conditions on a bounded domain with smooth boundary
    Specified for the model setup.

pith-pipeline@v0.9.0 · 5462 in / 1285 out tokens · 46680 ms · 2026-05-10T04:12:02.478031+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    Lotka, Elements of Physical Biology, Williams & Wilkins, New York, 1925

    A.J. Lotka, Elements of Physical Biology, Williams & Wilkins, New York, 1925

    work page 1925

  2. [2]

    Leonardo da Vinci

    V. Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, volume 2, Societa anonima tipografica “Leonardo da Vinci” (1927)

    work page 1927

  3. [3]

    Turing, The chemical basis of morphogenesis, Trans

    A.M. Turing, The chemical basis of morphogenesis, Trans. R. Soc. Lond., B237 (1952), 37-72

    work page 1952

  4. [4]

    Segel and J.L

    L.A. Segel and J.L. Jackson, Dissipative structure: an ecological example, J. Theor. Biol., 37 (1972), 545-559

    work page 1972

  5. [5]

    Chattopadhyay, P.K

    J. Chattopadhyay, P.K. Tapaswi, D. Datta, D. Chattopadhyay, Formation of a dissipative structure: A nonlinear analysis, Ecol. Model., 73 (1994), 205-214

    work page 1994

  6. [6]

    McLaughlin, J

    J.F. McLaughlin, J. Roughgarden, Predation across spatial scales in heterogeneous environments, Theor. Popul. Biol., 41 (1992), 277-299

    work page 1992

  7. [7]

    Ahmed, M.W

    N. Ahmed, M.W. Yasin, A. Akg¨ ul, D. Baleanu, O. Tintareanu-Mircea, Mathematical analysis and pattern formation in diffusive predator–prey system, J. Appl. Math. Comput., 71 (2025), 3037–3058. GLOBAL DYNAMICS AND PATTERNS IN A PREDATOR-PREY SYSTEM WITH TWO CHEMICALS 27

    work page 2025

  8. [8]

    L.N. Guin, S. Acharya, Dynamic behaviour of a reaction–diffusion predator–prey model with both refuge and harvesting, Nonlinear Dyn., 88 (2017), 1501–1533

    work page 2017

  9. [9]

    Gurtin, Some mathematical models for population dynamics that lead to segregation, Q

    M.E. Gurtin, Some mathematical models for population dynamics that lead to segregation, Q. J. Appl. Math., 32 (1974), 1-8

    work page 1974

  10. [10]

    G. Hu, X. Li, Y. Wang, Pattern formation and spatiotemporal chaos in a reaction–diffusion predator–prey system, Nonlinear Dyn., 81 (2015), 265–275

    work page 2015

  11. [11]

    Okubo, Diffusion and Ecological Problems: Mathematical Models

    A. Okubo, Diffusion and Ecological Problems: Mathematical Models. Biomathematics, Vol. 10. Springer, Berlin. (1980)

    work page 1980

  12. [12]

    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

    work page 2012

  13. [13]

    Tiwari, J.P

    V. Tiwari, J.P. Tripathi, D. Jana, S.K. Tiwari, R.K. Upadhyay, Exploring complex dynamics of spatial predator–prey system: role of predator interference and additional food, Int. J. Bifurcat. Chaos., 30 (2020), 2050102

    work page 2020

  14. [14]

    Raychaudhuri, D.K

    S. Raychaudhuri, D.K. Sinha, J. Chattopadhyay, Effect of time-varying cross-diffusivity in a two-species Lotka-Volterra competitive system, Ecol. Model., 92 (1996), 55-64

    work page 1996

  15. [15]

    Upadhyay, A

    R.K. Upadhyay, A. Patra, B. Dubey, N.K. Thakur, A predator–prey interaction model with self-and cross- diffusion in aquatic systems, J. Biol. Syst., 22 (2014), 1–22

    work page 2014

  16. [16]

    Ghorai, S

    S. Ghorai, S. Poria, Pattern formation and control of spatiotemporal chaos in a reaction diffusion prey–predator system supplying additional food, Chaos Solitons Fractals, 85 (2016), 57–67

    work page 2016

  17. [17]

    Q. Cao, J. Wu, Pattern formation of reaction–diffusion system with chemotaxis terms, Chaos, 31 (2021), 113118

    work page 2021

  18. [18]

    Mbopda, S

    B.T. Mbopda, S. Issa, S. Abdoulkary, R. Guiem, H.P. Fouda, Pattern formations in nonlinear dynamics of hepatitis B virus, Eur. Phys. J. Plus, 136 (2021), 586

    work page 2021

  19. [19]

    Ghorai, O

    S. Ghorai, O. Umut, S. Poria, Analysis of patterns in an additional food-provided predator–prey reaction diffusion model using amplitude equations, Pramana J. Phys., 97 (2023), 200

    work page 2023

  20. [20]

    Y. Wang, X. Zhou, W. Jiang, Bifurcations in a diffusive predator–prey system with linear harvesting, Chaos Solitons Fractals, 169 (2023), 113286

    work page 2023

  21. [21]

    Chakraborty, S

    B. Chakraborty, S. Marick, N. Bairagi, Diffusion-driven instabilities in a tri-trophic food web model: From Turing to non-Turing patterns and waves, Chaos Solitons Fractals, 189 (2024), 115634

    work page 2024

  22. [22]

    Mandal, L.N

    G. Mandal, L.N. Guin, S. Chakravarty, Complex patterns in a reaction–diffusion system with fear and anti- predator responses, Int. J. Bifurcat. Chaos., 34 (2024), 2450154

    work page 2024

  23. [23]

    S. Li, W. Jiang, X. Zhang, J. Wang, Dynamic analysis, patterns formation and numerical simulation of a reaction-diffusion system, Nonlinear Dyn., 113 (2025), 4923–4947

    work page 2025

  24. [24]

    Bhutia, S

    L.T. Bhutia, S. Biswas, E. Das, T.K. Kar, B. Bhunia, Evolution of Turing patterns of a predator–prey system with variable carrying capacity and harvesting, Chaos Solitons Fractals, 191 (2025), 115790

    work page 2025

  25. [25]

    Das, T.K

    E. Das, T.K. Kar, L.T. Bhutia, S. Biswas, B. Bhunia, Asymptotic and transient approaches of harvested predator–prey models with reaction–diffusion, Eur. Phys. J. Plus, 140 (2025), 310

    work page 2025

  26. [26]

    Mandal, L.N

    G. Mandal, L.N. Guin, S. Chakravarty, Cross-diffusion-induced instabilities in a cooperative hunting popu- lation with Allee effect, Eur. Phys. J. Plus, 140 (2025), 96

    work page 2025

  27. [27]

    P.J. Pal, D. Biswas, T. Saha, Spatial dynamics and pattern formation in fragmented habitats: A study using a diffusive Bazykin model with Allee effect, Chaos Solitons Fractals, 192 (2025), 116043

    work page 2025

  28. [28]

    Yasin, N

    M.W. Yasin, N. Ahmed, A. Akg¨ ul, M.Z. Baber, D. Baleanu, O. Tintareanu-Mircea, Spatio-temporal patterns and Turing–Hopf bifurcation in a spatially extended prey–predator model with ratio-dependent interactions, Model. Earth Syst. Environ., 11 (2025), 219

    work page 2025

  29. [29]

    Majee, S

    S. Majee, S. Jana, J. Ramprabhakar, Complex spatiotemporal dynamics in an eco-epidemic model: Turing instability, non-stationary patterns and chaos control, Math. Comput. Simul., 240 (2026), 650–667

    work page 2026

  30. [30]

    Murray, Mathematical Biology I: An Introduction, Springer–Verlag, New York, 2002

    J.D. Murray, Mathematical Biology I: An Introduction, Springer–Verlag, New York, 2002

    work page 2002

  31. [31]

    Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer–Verlag, New York, 2003

    J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer–Verlag, New York, 2003

    work page 2003

  32. [32]

    Abedon, Bacteriophage Ecology: Population Growth, Evolution and Impact of Bacterial Viruses, Cam- bridge University Press, 2009

    S.T. Abedon, Bacteriophage Ecology: Population Growth, Evolution and Impact of Bacterial Viruses, Cam- bridge University Press, 2009

    work page 2009

  33. [33]

    Weitz, T

    J.S. Weitz, T. Poisot, J.R. Meyer, C.O. Flores, S. Valverde, M.B. Sullivan, M.E. Hochberg, Phage–bacteria infection networks, Trends Microbiol., 21 (2013), 82–91

    work page 2013

  34. [34]

    Thornton, Dissolved organic matter (DOM) release by phytoplankton in the contemporary and future ocean, Eur

    D.C.O. Thornton, Dissolved organic matter (DOM) release by phytoplankton in the contemporary and future ocean, Eur. J. Phycol., 49 (2014), 20–46. 28 S. GNANASEKARAN, J. SAHA, O. D. MAKINDE, AND J. CHATTOPADHYAY

    work page 2014

  35. [35]

    Jiang, O.M.E

    L. Jiang, O.M.E. Schofield, P.G. Falkowski, Adaptive evolution of phytoplankton cell size, Am. Nat., 166 (2005), 496-505

    work page 2005

  36. [36]

    Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, 133 (1993), 9–126

    H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, 133 (1993), 9–126

    work page 1993

  37. [37]

    Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J

    J. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Differ. Equ., 259 (2015), 120–140

    work page 2015

  38. [38]

    Gnanasekaran, N

    S. Gnanasekaran, N. Nithyadevi, C. Udhayashankar, Global existence and asymptotic behavior of a preda- tor–prey chemotaxis system with inter-species interaction coefficients, J. Differ. Equ., 378 (2024), 264-302

    work page 2024

  39. [39]

    Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J

    X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181–188

    work page 2014

  40. [40]

    Hieber, J

    M. Hieber, J. Pr¨ uss, Heat kernels and maximalL q(Ω)-Lp(Ω) estimate for parabolic evolution equations, Commun. Partial Differ. Equ., 22 (1997), 1647–1669

    work page 1997

  41. [41]

    Ladyzenskaja, V

    O. Ladyzenskaja, V. Solonnikov, N. Uralceva, Linear and quasi-linear equations of parabolic type, American Mathematical Society, 1968

    work page 1968

  42. [42]

    Amann, Linear and Quasilinear Parabolic Problems: Volume I, Abstract Linear Theory, Birkh¨ auser Verlag, Basel, (1995)

    H. Amann, Linear and Quasilinear Parabolic Problems: Volume I, Abstract Linear Theory, Birkh¨ auser Verlag, Basel, (1995)

    work page 1995

  43. [43]

    Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715

    work page 2012

  44. [44]

    X. Bai, M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive ki- netics, Indiana Univ. Math. J., 65 (2016), 553-583

    work page 2016

  45. [45]

    Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J

    M. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483–508

    work page 2016

  46. [46]

    J. Wang, M. Wang, The diffusive Beddington–DeAngelis predator–prey model with nonlinear prey-taxis and free boundary, Math. Methods Appl. Sci., 41 (2018), 6741–6762

    work page 2018

  47. [47]

    T. Li, A. Suen, M. Winkler, C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721–746

    work page 2015

  48. [48]

    Porzio, V

    M.M. Porzio, V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103 (1993), 146–178. (GS)Department of Mathematics, National Institute of Technology Tiruchirappalli, Tamil- nadu 620015, India Email address:sekaran@nitt.edu (JS)Department of Mathematics, National Institute of Technology...

    work page 1993