Ecological dynamics of pro-tumor and anti-tumor teams in the tumor microenvironment

Cancer is a systemic disease in which cells escape the normal constraints of growth and regulation. Among its classical hallmarks [1] are sustained proliferative signaling and the evasion of growth suppressors, which together render tumor cells quasi-autonomous entities capable of unchecked expansion. This growth unfolds within a highly complex environment composed of diverse cell types that collectively form the tumor microenvironment (TME). Non-malignant cells within this environment were once thought to play a passive role in tumorigenesis; it has now become evident that they actively shape tumor progression, collectively forming a structured ecological community [1–6]. Although the evolutionary dynamics of cancer cells have been studied extensively [7–11], how these dynamics unfold on shorter time scales through ecological interactions within the TME remains underexplored [12–15]. Within the TME, several kinds of ecological interactions occur among cancer cells, immune cells, and stromal components. Cancer cells are subject to immune surveillance and also compete with host cells for limited resources such as oxygen, nutrients, and space. Recent work has established the clinical significance of tumor-immune ecology in the TME [16, 17], highlighting the need for a quantitative understanding of these ecological interactions. Early theoretical work introduced Lotka–Volterra-like equations to describe interactions between tumor cells and immune effectors [18], capturing phenomena such as tumor dormancy and immune-mediated control. Subsequent models expanded this framework to incorporate additional biological mechanisms, including cytokine dynamics and adoptive immunotherapy [19]. More recent studies have continued in this tradition, modeling tumor-host interactions as generalized Lotka–Volterra (GLV) systems to probe a broader range of ecological outcomes [20]. These models, however, remain largely low-dimensional and mechanistic, centering on a small number of interacting populations and therefore unable to capture the full heterogeneity of tumor ecology. A major challenge in modeling any ecosystem lies in its high dimensionality and complexity, which makes quantitative analysis difficult. Yet studies of ecological networks suggest that this apparent complexity often masks simpler underlying structure. Many real ecosystems can be decomposed into only a few functional modules [21], and their effective dimensionality is frequently much lower than expected [22]. Bipartite mutualistic networks provide a clear example of such structured ecological architecture, and have been characterized in detail [23, 24].

The ecological interactions in the TME are similarly not random and unstructured. Reviews of mathematical models of TME dynamics reveal that, despite differences in approach, technique, and the specific components modeled, most descriptions converge on interactions between pro- and anti-tumor factors [25]. The summary interaction network constructed by [25]—reproduced in figure 1 exhibits a clear two-team structure. Indeed, the immune and stromal cell types in the TME can be broadly grouped into two functional communities: an anti-tumor group arising from the adaptive and innate immune responses that target cancer cells, and a pro-tumor group typically driven by tumor-derived signals that recruit and reprogram inflammatory or immunosuppressive cells to secrete growth factors that support tumor expansion [3, 5, 26]. For instance, macrophages exhibit two phenotypes, M1 (anti-tumor) and M2 (pro-tumor) [27]; tumor-associated neutrophils exist in two distinct phenotypes, N1 (anti-tumor) and N2 (pro-tumor) [28]; and effector T cells (Teffs; anti-tumor) are functionally separated from regulatory T cells (Tregs; pro-tumor) [29, 30]. Cell types within each group tend to cooperate with others in the same group and antagonize those in the opposing group through mechanisms such as recruitment and activation of team members (see figure 1), implying an ecological system composed of two competing teams [25]. The precise dynamical regimes accessible to such a two-team system, however, remain unknown. While the Lotka–Volterra model has previously been adapted to study tumor-immune interactions, these models have primarily been mechanistic, capturing only a few components of the TME at once [18, 31], and further modifications adding more interaction pathways [19] have not resolved the fundamental challenge posed by the heterogeneity and complexity of the TME. The GLV equations offer a natural mathematical framework for analyzing such species-rich complex ecosystems [32, 33]. These models have been widely studied using the cavity method, which enables the determination of equilibrium statistics in the limit of large system size by treating interaction parameters as drawn from statistical ensembles [34–38]. Importantly, recent work [39] has shown that many macroscopic properties of complex ecosystems are largely insensitive to microscopic details and can be captured by a small number of statistical parameters describing a GLV model, with deviations accounted for by introducing structure into the interaction matrix [39]. In our case, this structure is motivated biologically by the organization of the TME into pro-tumor and anti-tumor functional groups and we study a structured two-team ecosystem driven by GLV dynamics, with interaction parameters drawn from Gaussian ensembles whose variances encode ecological heterogeneity. Using the cavity method, we derive equilibrium statistics, including survival fractions and mean abundances for both teams. By tuning the balance between intra-team cooperation and inter-team inhibition, we uncover transitions between mutual coexistence, competitive exclusion of one team. We further show that heterogeneity in interaction strengths within and across teams shapes these transitions in distinct ways. Taken together, these results provide a statistical physics based ecological framework for understanding the collective dynamics between pro-tumor and anti-tumor factors in the TME.

Figure 1. Teams in the tumor microenvironment. (A) Network representation of interactions among major cell types in the TME, reveals two distinct communities: the anti-tumor team (green) and the pro-tumor team (red). (B) Adjacency matrix illustrating activating and inhibitory interactions between cell types.

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As mentioned, we model pro-tumor and anti-tumor cell types as two interacting teams of species, whose dynamics are governed by the GLV equations. In the simplest unstructured community setting, the abundance $N_i$ of species $i$ evolves according to

Equation or symbol description not available

where $N_i$ denotes the normalized abundance of species $i$ (scaled by its carrying capacity) and $S$ is the total number of species in the community. Here, $A_$ represents the effect of species $j$ on $i$, $A_ \lt 0$ indicates inhibition of $i$ by $j$, and $A_ \gt 0 $ indicates activation of $i$ from $j$. In the rescaled units, $r_$ functions as an effective carrying capacity and is drawn from a normal distribution $r_i \sim \mathcal(\mu_r, \zeta^2)$.

In the cavity method framework, the interaction coefficients are defined as

Equation or symbol description not available

where $ \langle a_ a_ \rangle = \gamma $. For simplicity, we will mostly focus on the case $\gamma = 0$, but non-zero $\gamma$ results will be presented in the supplementary figures. The $1/S$ and $1/\sqrt$ scaling of the mean and variance of $A_$ ensures a well-defined thermodynamic limit as the number of species $S$ becomes large. Specifically, the $1/S$ factor normalizes the cumulative contribution of all interactions so that the mean interaction effect on each species remains $\mathcal(1)$, while the $1/\sqrt$ factor guarantees that the fluctuations arising from random interactions also remain finite. This scaling allows the macroscopic properties of the system-such as species survival fractions, mean abundances, and stability-to converge to deterministic values as $S \to \infty$.

We can then derive precise statistics of a random community ($\phi, \langle N \rangle, \langle N^2 \rangle) $ for a given $\mu$ and $\sigma$ defining the dynamical ensemble. In unstructured random communities, when the heterogeneity ($\sigma$) is above a certain threshold, we enter a disordered regime of chaotic dynamics (figures 2(A) and (C)) [34, 35]. Otherwise, we have steady-state coexistence of various species, as can be seen in simulations of specific ecosystem realizations in this parameter range (figure 2(B)).

Figure 2. High heterogeneity leads to extinctions in ecosystems with teams (A) Survival fraction for an unstructured random ecosystem in the $(\sigma,\mu)$ plane ($\gamma = 0$). The dashed red line marks the onset of the multiple-attractor phase; red shading indicates cavity-method non-convergence. (B) Dynamics of a 20-species GLV system sampled from the unique-fixed-point phase ($\mu = -1$, $\sigma = 0.2$) and ($\mu_ = 1, \zeta = 0$) in the unstructured random ecosystem showing stable convergence. (C) The same system sampled at high heterogeneity ($\mu = -1$, $\sigma = 1.6$), showing unbounded trajectories. (D) Phase diagram for Pro-tumor team in a two team structured ecosystem in $(\mu_},\sigma^})$, with $\mu^ = \mu^ = \mu_}$ and $\mu^ = \mu^ = -\mu_}$, $\zeta^ = \zeta^A = 0, \quad \mu_^ = \mu_^ = 1$. Increasing $\mu_}$ strengthens both intra-team cooperation and inter-team antagonism. (E) GLV simulation of two 20-species teams at $\sigma^} = 0.2$, $\mu^ = \mu^ = 0.5$, $\mu^ = \mu^ = -1$, showing stable coexistence. (F) The same system at $\sigma^} = 1.6$, showing unbounded growth.

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We now generalize the method to a structured ecosystem with two teams, a pro-tumor team and an anti-tumor team. Each species belongs to exactly one team, and team membership is fixed throughout. The dynamics of the $i}$ species belonging to the pro-tumor team (pro-tumor team denoted by T and anti-tumor team denoted by A) is given by

Equation or symbol description not available

and that of $i}$ species belonging to anti-tumor team are given by

Equation or symbol description not available

where $r_i^} \sim \mathcal(\mu_r^}, (\zeta^})^)$ (for $y = \$) represents the group-specific effective carrying capacity distribution. Each group $T/A$ consists of $S^$ species, and the total number of species in the ecosystem is $S = S^T + S^$. The interaction matrix between species belonging to groups $ T $ and $ A $ is defined as

Equation or symbol description not available

with $a_^ \sim \mathcal(0,1)$. $A_ ^, A_ ^, A_ ^$ are defined similarly. We denote the pro-tumor team by $T$ and anti-tumor team by $A$ and therefore $\mu^, \mu^, \mu^, \mu^$ are the means for the interactions within the pro-tumor team, within the anti-tumor team, on the anti-tumor team from the pro-tumor team and from the anti-tumor team on the pro-tumor team respectively.

Extending the cavity method to this case (see appendix), we are able to solve for survival fraction $\phi^$, first moment/mean population $ \langle N^ \rangle $ and second moment $ \langle (N^)^ \rangle $ for each group given the interaction means $\mu^, \mu^, \mu^,\mu^,$ and standard deviations $ \sigma^, \sigma^, \sigma^, \sigma^ $, as well as the effective carrying capacity standard deviation $\zeta^, \zeta^$. Each team occupies a fraction $\rho^ = \frac}$ of the total community, allowing us to analyze inter-team and intra-team interactions in a tractable manner.

To begin to contrast the pro-tumor/anti-tumor team ecosystem with an unstructured random community, we define a composite parameter $\mu_}$ which captures the balance between cooperation (within members belonging to same team) and inhibition (among members belonging to separate teams).

Equation or symbol description not available

and define $\sigma^} = \sigma^ = \sigma^ = \sigma^ = \sigma^$ We have a two-team ecosystem when $\mu_} \gt 0$, and larger values indicate stronger competition between the two teams with stronger cooperation within the teams. On the other hand, $\mu_} \lt 0$ denotes the case where there is antagonism within the teams but cooperation between species in different teams. This may arise in a system with competition among related species for the benefits arising from mutualistic interactions with other parts of the ecosystem.

More specifically, in fully random ecosystems, the sign of the mean interaction strength sets the overall ecological ‘mode’: $\mu \gt 0$ corresponds to net cooperation, while $\mu \lt 0$ corresponds to net competition. In our two-team ecosystem with positive $\mu_}$, however, the situation is more nuanced-cooperative interactions occur within each community, while competitive interactions act between them. Despite this structural difference, both random ecosystems and the two-team ecosystem display a common trend: increasing interaction heterogeneity drives species extinctions.

As we have seen, in random ecosystems once the variance of interactions exceeds a critical threshold, the system transitions from a regime with stable abundances to a phase with fluctuating abundances. Simulations in the former regime produce clean, stable convergence for a 20-species community (figure 2(B)), whereas simulations in the multiple-attractor regime show highly variable non-steady outcomes (figure 2(C)). The red region in figure 2(A) marks where the cavity equations do not converge. This typically occurs when cooperation dominates ($\mu \gt 0$), but sufficiently high heterogeneity can also destabilize systems that are, on average, competitive ($\mu \lt 0$). Importantly, the amount of heterogeneity that leads to non-convergence increases as baseline competition becomes stronger.

The two-team ecosystem behaves differently. Here, ‘competition balances cooperation’, namely strong antagonistic interactions between teams stabilize the system, and convergence persists even at high values of $\mu_}$. We find that cavity-solver non-convergence in the two-team model occurs only at higher heterogeneity. (approximately $\sigma \gt 1.0$). For $\sigma^} \lt 1$, both teams robustly coexist, consistent with direct simulations of finite ecosystems with 20 species per team (figure 2(E)). In the two-team ecosystem, we reach cavity non-convergence at $\sigma^} \gt 1$, and simulations of several iterates in this regime exhibit diverging abundances (figure 2(F))

2.2. An ecosystem with two competing teams results in distinct regimes of coexistence and exclusion

So far, we have examined the role of a single parameter $\mu_}$ that enforced identical intra-community cooperation. However, the two teams could have different strengths of cooperation and competition. Here, we generalize this analysis by independently varying $\mu^$ and $\mu^$. Throughout this section, we fix $\sigma^} = 0.2$ and assume that both communities are of equal size ($\rho^ = \rho^ = 0.5$). The inter-team interactions are taken to be inhibitory, with $\mu^ = \mu^ = -1$, reflecting the antagonistic nature of the tumor-versus-anti-tumor interplay.

For each pair $(\mu^, \mu^)$, we solve the cavity equations self-consistently to obtain the survival fractions and mean population levels of the two communities, namely $\phi^$, $\phi^$, $\langle N^ \rangle$, and $\langle N^ \rangle$. A two team ecosystem with intra-team cooperation and inter-team antagonism exists when $\mu^, \mu^ \gt 0 $

As seen previously in figure 2(E), species from both teams coexist with comparable population sizes when the teams are identical with $\mu^ = \mu^ = 0.5$. In the more general $(\mu^, \mu^)$ plane, we observe coexistence between the tumor and anti-tumor communities whenever $\mu^, \mu^ \unicode 1$. This regime includes cases with inhibitory intra-team interactions ($\mu^, \mu^ \unicode 0$), yet both teams maintain nonzero survival fractions (figures 3(A) and (B)).

Figure 3. Phase diagrams of an ecosystem with two teams. Self-consistent equations for tumor and anti-tumor teams solved in the $(\mu^, \mu^)$ space under inhibitory interactions ($\mu^ = \mu^ = -1$) with heterogeneity ($\sigma^ = \sigma^ = \sigma^ = \sigma^ = 0.2$,

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PHYSICAL BIOLOGY

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