Abstract
Competitive interactions between multiple cell populations are crucial for modeling biological processes such as tumor growth and tissue regeneration. In this study, we investigate the nonlocal advection model for two-species competition, which characterizes cell growth and dispersion phenomena in co-culture experiments. To capture realistic phenomena in biology, we introduce the time delay representing population migration and resources recovering time. The primary objectives are to investigate the impact of time delays on competitive dynamics under various parameter settings and to develop numerical methods that ensure biological feasibility of solutions. Accordingly, we design a positivity-preserving finite volume scheme based on an upwind flux approach, guaranteeing non-negative population densities and discrete conservation properties. We examine the convergence orders of the scheme through the numerical experiments and explore the effects of time delays on species competition dynamics under different parameter settings.
1 Introduction
The mathematical modeling of multi-species population dynamics provides a theoretical framework for studying species interactions in biological and ecological systems [1, 2]. In biochemical research, scientists employ co-culture systems to elucidate competitive relationships between distinct cell types, such as cancer cells and normal cells [3]. Early models included nonlinear diffusion and reaction-diffusion systems [4, 5, 6], revealing that spatial segregation can promote species coexistence by mitigating interspecific competition. A key approach in modern models involves nonlocal advection systems [7, 8], where population movement is influenced by long-range interactions instead of purely local diffusion. These models are often associated with chemotaxis, the directed movement of cells or organisms along chemical concentration gradients [9, 10, 11, 12]. The well-posedness of nonlocal advection models for two species (or a single species) has been studied in [7, 8, 13, 14]. Moreover, nonlocal advection systems are closely connected to continuum models of pedestrian dynamics, where an individual’s velocity is influenced not only by local density but also by spatially distributed crowd information [15, 16]. Extensions considering anisotropic interactions and domain boundaries further demonstrate the relevance of nonlocal frameworks to realistic crowd flow modeling [17, 18].
Time delays capture the inherent non-instantaneous nature of biological processes, yielding more realistic models that can exhibit oscillatory, bifurcation, and stability behaviors observed in real systems [19, 20]. Following the pioneering work of [21], numerous studies have introduced time delays in population dynamics models. In single species models such as the delayed logistic model [22, 23, 24], the delay represents the maturation time before individuals become reproductively active and can destabilize population dynamics by inducing oscillations or periodic behavior once it exceeds a critical threshold. The time delayed Lotka–Volterra predator-prey model [25, 23, 26] incorporates the predator’s delayed response to changes in prey population, enabling more accurate simulations of periodic oscillations and stability shifts in ecosystems. Time lags in competition models [27, 28] physically represent ecological processes such as resource recovery or interspecific interaction lags, and they can destabilize equilibrium states to induce stable periodic oscillations or even chaotic dynamics through Hopf bifurcation or period-doubling bifurcations.
This study aims to extend the existing nonlocal advection model for two competing species by incorporating time delays. Previous studies focused on population mobility and interactions, while our study introduces time delays to characterize the interaction between population migration and resource recovery. To ensure biologically realistic population densities, we develop a positivity-preserving finite volume scheme. We conduct extensive numerical experiments to: (1) examine the numerical convergence orders; (2) investigate the influence of time delays on competitive dynamics under various parameter settings.
In this work, we first introduce the mathematical model—the two species hyperbolic Keller–Segel system with time delays, and then design a finite volume scheme satisfying the positivity-preserving property, where we adopt the upwind type flux similar to [29–32, 33]. We investigate the experimental convergence orders of the proposed scheme. According to the numerical simulation, under various parameter settings, we find that small, identical delays in both species lead to a steady state, whereas larger delays may induce unsteady and potentially the extinction of one or both species. Moreover, asymmetric delay parameters between the two species tend to confer a competitive advantage to one species.
The rest of this paper is organized as follows. In Section 2, we extend the nonlocal advection Keller–Segel model by incorporating time delay and develop a positivity-preserving finite volume method that also satisfies discrete conservation laws. Numerical examples are presented in Section 3.
2 The mathematical model and the finite volume scheme
In this section, we introduce the time delay to the classical hyperbolic Keller–Segel system and design a positivity-preserving finite volume scheme.
2.1 PDE model
We consider the nonlocal advection system for two competing species with time delays in an interval . We denote by the density of two species. The PDE model is stated as follows.
where is the outward normal vector, is the dispersion coefficient, is the sensing coefficient. The function , which represents the pressure or chemical potential field induced by species densities, satisfies the elliptic Equations 2.1c, 2.1d, and its gradient influences the movement direction of the species . We set, for ,
where are the growth rates, represent the intraspecific competition coefficients, reflecting competition between individuals of the same species, denote the interspecific competition coefficients between species, represent the delay parameters, and are the additional mortality rates caused by drug treatment. The delay parameters in represent time lags in intraspecific competition of species , reflecting delayed self-density feedback from migration and resource recovery. When , Equations 2.1a–2.1f is the hyperbolic Keller–Segel system proposed in [14].
2.2 The finite volume scheme
We introduce the fully discrete finite volume scheme for problem Equations 2.1a–2.1f. We hereafter denote by the measure of the interval in .
2.2.1 The finite volume scheme
For simplicity, we divide the computational domain into cells (see Figure 1):whereWe denote by the center of the cell , and setWe introduce the space of the piecewise constant function:where represents the set of constant functions in cell .
FIGURE 1
We assume that there are two integers and such that the delay parameters satisfy . Let for some integer K. We can take the integers , . Then we see that , .
Let be the smallest integer such that . The time interval is divided into segments:and we use the backward Euler to approximate time-differential, i.e.,
Let be the numerical approximation to . The proposed scheme is described below.
By integrating Equations 2.1a–2.1c on each cell and applying integration by parts, we obtain the following implicit scheme at using the backward Euler method:Using on , we get:We use the approximation:
(by Equation 2.1d)
We apply the upwind discretization to treat the flux: for ,where and . The upwind type numerical flux can be understood as follows. If the “flux” is from towards (i.e., ), then the numerical flux for is chosen as ; otherwise, we take .
For simplicity, we set the notations:The time delay terms are approximated bywhere the integers satisfy
For , by the fact that , we get the finite volume scheme (FVM) scheme.
2.2.2 Positivity-preserving property
In this section, we show the positivity-preserving for the discrete solutions. We set .
Assume that(,) are nonnegative and not identically zero. Ifis sufficiently small such that, then we have:
Proof. The discrete system Equations 2.4b, 2.4c can be rewritten into the matrix forms:where ,
We assume that (, ). In view of Equations 2.2a, 2.2b, we see that . If we choose , then we obtain:
The following statements hold for
.
1. has positive diagonal entries, i.e., for and ,
2. has non-positive off-diagonal entries, i.e., for and ,
3. All the column sums of are positive, i.e., for and ,
According to a convenient sufficient but not necessary condition of M-matrices (cf. [
5, Appendix]), we conclude that
is an M-matrix.
According to the properties of M-matrices, this implies that . Noting that (, ), we have . By induction, we conclude that for all .
2.2.3 The discrete conservation laws
The purpose of this section is to establish the mass conservation property of the (FVM) scheme (see Equations 2.4a–2.4c).
Letbe the solution of (FVM). Then, we have the discrete conservation law:Moreover, under the assumptions that , it follows that
Proof. Summing up Equations 2.4b, 2.4c with respect to , we obtainWe see thatwhere
(by Equation 2.3).
Hence, we haveMoreover, if we assume that (i.e., ), then we haveHence the proof is complete.
3 Numerical experiments
In this section, we conduct extensive numerical experiments, where we examine the numerical convergence orders, compare the results with the classical hyperbolic Keller–Segel system, and investigate the influence of time delays on competitive dynamics under various parameter settings. We take in this section. We set.
3.1 Example 1 (non-segregated initial functions)
We set
,
,
,
, and
The initial conditions are chosen as follows:
To examine the influence of time delays on competitive dynamics under varying parameter settings, the following cases are considered.
1. the classical hyperbolic Keller–Segel system
Example 1-1: ;
2.
Example 1-2-1: ;
Example 1-2-2: ;
Example 1-2-3: ;
Example 1-2-4: ;
Example 1-2-5: ;
3.
Example 1-2-6: ;
Example 1-2-7: ;
Example 1-2-8: ;
Example 1-2-9: .
To obtain the experimental convergence order of the mesh size , we fix , use the numerical solution with at as the reference solution (denoted by (, , )), and compute the errors for different mesh sizes . For the classical hyperbolic Keller–Segel system (i.e., ), the experimental errors are shown in Table 1. For the system with time delay , the experimental errors are presented in Table 2. We observe the first-order convergence with respect to the mesh size for and .
TABLE 1
| h | rate | rate | rate | |||
|---|---|---|---|---|---|---|
| 1/80 | 1.29E-01 | - | 3.02E-02 | - | 1.11E-02 | - |
| 1/160 | 8.23E-02 | 0.64 | 1.78E-02 | 0.76 | 5.90E-03 | 0.90 |
| 1/320 | 4.81E-02 | 0.78 | 9.80E-03 | 0.86 | 3.00E-03 | 1.01 |
| 1/640 | 2.38E-02 | 1.01 | 4.70E-03 | 1.07 | 1.30E-03 | 1.21 |
Experimental errors and convergence order of for (Example 1-1).
TABLE 2
| h | rate | rate | rate | |||
|---|---|---|---|---|---|---|
| 1/80 | 1.35E-01 | - | 3.25E-02 | - | 1.20E-02 | - |
| 1/160 | 8.55E-02 | 0.66 | 1.89E-02 | 0.78 | 6.40E-03 | 0.90 |
| 1/320 | 4.95E-02 | 0.79 | 1.03E-02 | 0.88 | 3.20E-03 | 1.02 |
| 1/640 | 2.43E-02 | 1.03 | 4.80E-03 | 1.08 | 1.40E-03 | 1.22 |
Experimental errors and convergence order of for (Example 1-2-2).
The evolution of the solutions for the classical hyperbolic Keller–Segel system (i.e., ) with is shown in Figure 2. The solution dynamics of the time-delayed nonlocal advection model with are presented in Figures 3–6. We classify the dynamic behaviors for and in Tables 3, 4, respectively. For the non-segregated initial function cases, small symmetric delays lead to steady coexistence (see Example 1-2-1 – Example 1-2-2, Figure 3), while larger delays cause unsteady dynamics and potential extinction of one or both species (see Example 1-2-3 – Example 1-2-5, Figure 4). When the time delay is identical for both species (i.e., ) and sufficiently large, the delayed competitive feedback leads to an unsteady system. The populations overreact to outdated density information, causing oscillations that eventually drive both species to extinction, as seen in the numerical results. For the case with asymmetric delays, we observe that the species with the larger delay is dominance in the competition (e.g., dominates when , or vice versa (see Example 1-2-6 – Example 1-2-9, Figures 5, 6)).
FIGURE 2
FIGURE 3
FIGURE 4
FIGURE 5
FIGURE 6
TABLE 3
| Example | Steady state | Long time behavior | |
|---|---|---|---|
| Example 1-2-1 | 0.1 | Steady state | Both non-vanishing and steady |
| Example 1-2-2 | 0.5 | Steady state | Both non-vanishing and steady |
| Example 1-2-3 | 1.5 | No steady state | Both non-vanishing and unsteady |
| Example 1-2-4 | 2 | No steady state | One vanished and the other one is unsteady |
| Example 1-2-5 | 5 | No steady state | Both vanished |
Classification of asymptotic behavior .
TABLE 4
| Example | Long time behavior | ||
|---|---|---|---|
| Example 1-2-6 | 0.5 | 1 | Steady and superior |
| Example 1-2-7 | 1 | 0.5 | Steady and superior |
| Example 1-2-8 | 0.5 | 1.5 | Steady and superior |
| Example 1-2-9 | 1.5 | 0.5 | Steady and superior |
Classification of asymptotic behavior .
3.2 Example 2 (segregated initial functions)
Set
,
,
,
, and
The initial conditions are set to
We consider the following settings.
1. the classical hyperbolic Keller–Segel system
Example 2-1: ;
2.
Example 2-2-1: ,
Example 2-2-2: ,
Example 2-2-3: ,
Example 2-2-4: ,
Example 2-2-5: ,
Example 2-2-6: ;
3.
Example 2-2-7: ,
Example 2-2-8: .
We fix , take the numerical solution with at as the reference solution, and compute the errors for mesh sizes . The experimental errors for and , shown in Tables 5, 6, respectively, indicate first-order convergence with respect to the mesh size for and .
TABLE 5
| h | rate | rate | rate | |||
|---|---|---|---|---|---|---|
| 1/80 | 6.10E-02 | - | 6.10E-02 | - | 2.24E-02 | - |
| 1/160 | 3.53E-02 | 0.79 | 3.56E-02 | 0.78 | 1.13E-02 | 0.99 |
| 1/320 | 1.95E-02 | 0.86 | 1.97E-02 | 0.85 | 5.37E-03 | 1.07 |
| 1/640 | 9.58E-03 | 1.02 | 9.70E-03 | 1.02 | 2.25E-03 | 1.26 |
Experimental errors and convergence order of for (Example 2-1).
TABLE 6
| h | rate | rate | rate | |||
|---|---|---|---|---|---|---|
| 1/80 | 6.14E-02 | - | 6.14E-02 | - | 2.28E-02 | - |
| 1/160 | 3.54E-02 | 0.79 | 3.58E-02 | 0.78 | 1.14E-02 | 1.00 |
| 1/320 | 1.94E-02 | 0.87 | 1.97E-02 | 0.86 | 5.40E-03 | 1.08 |
| 1/640 | 9.50E-03 | 1.04 | 9.60E-03 | 1.03 | 2.20E-03 | 1.27 |
Experimental errors and convergence order of for (Example 2-2-2).
Figure 7 displays the evolution of solutions for the classical Keller–Segel system with . Figures 8–10 illustrate the dynamic behavior of the time-delayed nonlocal advection model with . Tables 7, 8 summarize the dynamic behaviors corresponding to the cases and , respectively. Symmetric delays in segregated initial functions, where the coefficients for and are the same, demonstrate that small delays can support steady coexistence. However, larger delays may result in unsteady states or extinction. In Figure 10, we consider the case where , revealing that the species with the larger delay often gain a competitive advantage.
FIGURE 7
FIGURE 8
FIGURE 9
FIGURE 10
TABLE 7
| Example | Steady state | Long time behavior | |
|---|---|---|---|
| Example 2-2-1 | 0.1 | Steady state | Both non-vanishing and steady |
| Example 2-2-2 | 0.5 | Steady state | Both non-vanishing and steady |
| Example 2-2-3 | 1 | No steady state | Both non-vanishing and unsteady |
| Example 2-2-4 | 1.5 | No steady state | One vanished and the other one is unsteady |
| Example 2-2-5 | 2 | No steady state | Both vanished |
| Example 2-2-6 | 2.5 | No steady state | Both vanished |
Classification of asymptotic behavior .
TABLE 8
| Example | Long time behavior | ||
|---|---|---|---|
| Example 2-2-7 | 0.5 | 1 | Unsteady and superior |
| Example 2-2-8 | 1 | 0.5 | Unsteady and superior |
Classification of asymptotic behavior .
3.3 Example 3 (segregated initial functions)
The settings are the same as those in Example 2, except that we choose
The following parameters are considered.
1. the classical hyperbolic Keller–Segel system
Example 3-1: ;
2.
Example 3-2-1: ,
Example 3-2-2: ,
Example 3-2-3: ,
Example 3-2-4: ;
3.
Example 3-2-5: ,
Example 3-2-6: .
We fix , take the numerical solution with at as the reference solution, and compute the errors for mesh sizes . First-order convergence with respect to the mesh size is observed for and , as shown in Table 9 for , and Table 10 for .
TABLE 9
| h | rate | rate | rate | |||
|---|---|---|---|---|---|---|
| 1/80 | 6.28E-02 | - | 6.25E-02 | - | 2.12E-02 | - |
| 1/160 | 3.61E-02 | 0.80 | 3.69E-02 | 0.76 | 1.07E-02 | 0.99 |
| 1/320 | 1.99E-02 | 0.86 | 2.05E-02 | 0.84 | 5.12E-03 | 1.07 |
| 1/640 | 9.77E-03 | 1.03 | 1.02E-02 | 1.01 | 2.15E-03 | 1.25 |
Experimental errors and convergence order of for (Example 3-1).
TABLE 10
| h | rate | rate | rate | |||
|---|---|---|---|---|---|---|
| 1/80 | 6.46E-02 | - | 6.27E-02 | - | 2.20E-02 | - |
| 1/160 | 3.71E-02 | 0.80 | 3.68E-02 | 0.77 | 1.10E-02 | 0.99 |
| 1/320 | 2.04E-02 | 0.86 | 2.03E-02 | 0.86 | 5.20E-03 | 1.08 |
| 1/640 | 1.01E-02 | 1.02 | 1.00E-02 | 1.03 | 2.20E-03 | 1.27 |
Experimental errors and convergence order of for (Example 3-2-2).
The evolutions of the solution for the classical Keller–Segel system are shown in Figure 11. Figures 12, 13 present the evolutions of the solution for the nonlocal advection model with time delays . The dynamic behaviors for and are summarized in Tables 11, 12, respectively. For the symmetric delays in the segregated initial functions, where the coefficients for and differ, small delays still result in the steady state (see Example 3-2-1 – Example 3-2-2, Figure 12). On the other hand, larger delays can lead to unsteady dynamics and extinction (see Example 3-2-3 – Example 3-2-4, Figure 12). For the case with asymmetric delays, we see that both two examples are superior (see Example 3-2-5 – Example 3-2-6, Figure 13).
FIGURE 11
FIGURE 12
FIGURE 13
TABLE 11
| Example | Steady state | Long time behavior | |
|---|---|---|---|
| Example 3-2-1 | 0.1 | Steady state | One vanished and the other one is steady |
| Example 3-2-2 | 0.5 | Steady state | One vanished and the other one is steady |
| Example 3-2-3 | 1 | No steady state | One vanished and the other one is unsteady |
| Example 2-2-4 | 4 | No steady state | Both vanished |
Classification of asymptotic behavior .
TABLE 12
| Example | Long time behavior | ||
|---|---|---|---|
| Example 3-2-5 | 0.5 | 1 | Unsteady and superior |
| Example 3-2-6 | 1 | 0.5 | Steady and superior |
Classification of asymptotic behavior .
Remark 3.1We observe that small symmetric delays result in the steady state. This suggests that ecosystems with balanced feedback mechanisms (e.g., similar resource recovery times for competing species) are more likely to maintain biodiversity. On the other hand, larger symmetric delays can lead to unsteady dynamics and extinction, which align with scenarios where delayed interventions (e.g., pesticide application or vaccination campaigns) fail to prevent population collapse. Moreover, asymmetric delay parameters between the two species tend to confer a competitive advantage to one species. Such dynamics are relevant in tumor-microenvironment interactions, where targeting delay mechanisms could alter competitive outcomes.
4 Conclusion
This study proposes a nonlocal advection model with time delays to investigate the population dynamics of two competing species, extending previous frameworks by incorporating delayed interactions between migration and resource recovery. A positivity-preserving finite volume scheme is developed to ensure biologically realistic solutions. Numerical experiments demonstrate that small symmetric delays are steady, while larger delays induce unsteady states and potential extinction of one or both species. Moreover, asymmetric delay parameters between the two species tend to confer a competitive advantage to one species. Future work may explore applications to more complex ecological systems, incorporate stochastic or spatially heterogeneous delays, and extend the model to multi-species interactions or adaptive delay mechanisms for deeper insights into delayed feedback effects in biological processes. As a future work, we would like to consider the development of high-dimensional chemotaxis models with time delays, explore alternative numerical schemes to improve computational efficiency, and ensure that the system properties (such as positivity preservation, conservation laws, and energy dissipation) are maintained.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
PZ: Writing – original draft, Data curation, Writing – review and editing, Methodology. YE: Writing – review and editing, Funding acquisition, Supervision, Methodology. GZ: Funding acquisition, Methodology, Supervision, Investigation, Writing – review and editing.
Funding
The author(s) declare that financial support was received for the research and/or publication of this article. This work was supported by JSPS KAKENHI Grant Number 25K07137, NSFC General Project No. 12171071, and the Natural Science Foundation of Sichuan Province No. 2023NSFSC0055.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declare that no Generative AI was used in the creation of this manuscript.
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Summary
Keywords
two species model, time delay, hyperbolic PDE, Keller–Segel model, advection, nonlocal
Citation
Zeng P, Enatsu Y and Zhou G (2025) A nonlocal advection system for two competing species with resources recovering time. Front. Phys. 13:1657967. doi: 10.3389/fphy.2025.1657967
Received
02 July 2025
Accepted
19 September 2025
Published
03 November 2025
Volume
13 - 2025
Edited by
Ryosuke Yano, Tokio Marine dR Co., Ltd., Japan
Reviewed by
Abdellatif Ben Makhlouf, University of Sfax, Tunisia
Norikazu Saito, The University of Tokyo, Komaba, Japan
Updates
Copyright
© 2025 Zeng, Enatsu and Zhou.
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*Correspondence: Yoichi Enatsu, yenatsu@rs.tus.ac.jp
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