Evolutionary therapy (ET) applies principles of evolutionary biology to steer tumour dynamics and forestall or delay treatment resistance, typically guided by data-driven mathematical models. Our aim is to assess whether ET protocols, and specifically Zhang et al.’s protocol proposed for metastatic castrate-resistant prostate cancer, can be theoretically effective for fast-growing metastatic cancers such as stage IV non-small-cell lung cancer (NSCLC). Using longitudinal tumour-burden data from NSCLC patients treated with erlotinib, we systematically evaluate 26 two-population differential-equation models based on classical tumour-growth dynamics, with varying assumptions about density- and frequency-dependent interactions, pharmacokinetics, and treatment-induced death. Previous work by Yin et al. on the same dataset employed an exponential model that omitted density- and frequency-dependent interactions; although it provided a good fit to tumour-burden data, its structure would theoretically lead to poorer outcomes under ET protocols. In contrast, our analysis identifies the minimal model structure required to reproduce the resistance-driven regrowth observed in NSCLC, with the Gompertzian model featuring log-kill dynamics and both density- and frequency-dependent interactions providing the best fit. In this model, Zhang et al.’s protocol prolonged median time-to-progression to 42.3 months compared with 24.8 months under maximum tolerated dose. These results indicate that ET is theoretically a viable treatment strategy for NSCLC. This study offers a practical framework for assessing ET feasibility using clinical data and supports future clinical translation of ET in NSCLC.

In mathematical models of eco-evolutionary dynamics with a quantitative trait, two species with different strategies can coexist only if they are separated by a valley or peak of the adaptive landscape. A community is ecologically and evolutionarily stable if each species’ trait sits on global, equal fitness peaks, forming a saturated ESS community. However, the adaptive landscape may allow communities with fewer (undersaturated) or more (hypersaturated) species than the ESS. Non-ESS communities at ecological equilibrium exhibit invasion windows of strategies that can successfully invade. Hypersaturated communities can arise through mutual invasibility where each non-ESS species’ strategy lies in another’s invasion window. Hypersaturation in ESS communities with more than 1 species remains poorly understood. We use the G-function approach to model niche coevolution and Darwinian dynamics in a Lotka–Volterra competition model. We confirm that up to 2 species can coexist in a hypersaturated community with a single-species ESS if the strategy is scalar-valued, or 3 species if the strategy is bivariate. We conjecture that at most n·s+1 species can form a hypersaturated community, where n is the number of ESS species at the strategy’s dimension s. For a scalar-valued 2-species ESS, 4 species coexist by “straddling” the would-be ESS traits. When our model has a 5-species ESS, we can get 7 or 8, but not 9 or 10, species coexisting in the hypersaturated community. In a bivariate model with a single-species ESS, an infinite number of 3-species hypersaturated communities can exist. We offer conjectures and discuss their relevance to ecosystems that may be non-ESS due to invasive species, climate change, and human-altered landscapes.

Evolutionary cancer therapy (ECT) delays or forestalls the progression of metastatic cancer by adjusting treatment based on individual patient and disease characteristics. Clinical implementation of ECT can improve patient outcomes but faces technical and cultural challenges. To address those, we propose a systems approach incorporating systems modeling, problem structuring, and stakeholder engagement. This approach identifies and addresses barriers to implementation, ensuring the feasibility of ECT in clinical practice and enabling better metastatic cancer care.

We analyze the stability of a game-theoretic model of a polymorphic eco-evolutionary system in the presence of human intervention. The goal is to understand how the intensity of this human intervention and competition within the system impact its stability, with cancer treatment as a case study. In this case study, the physician applies anti-cancer treatment, while cancer, consisting of treatment-sensitive and treatment-resistant cancer cells, responds by evolving more or less treatment-induced resistance, according to Darwinian evolution. We analyze how the existence and stability of the cancer eco-evolutionary equilibria depend on the treatment dose and rate of competition between cancer cells of the two different types. We also identify initial conditions for which the resistance grows unbounded. In addition, we adopt the level-set method to find viscosity solutions of the corresponding Hamilton–Jacobi equation to estimate the basins of attraction of the found eco-evolutionary equilibria and simulate typical eco-evolutionary dynamics of cancer within and outside these estimated basins. While we illustrate our results on the cancer treatment case study, they can be generalized to any situation where a human aims at containing, eradicating, or saving Darwinian systems, such as in managing antimicrobial resistance, fisheries management, and pest management. The obtained results help our understanding of the impact of human interventions and intraspecific competition on the possibility of containing, eradicating, or saving evolving species. This will help us with our ability to control such systems.

We present a game-theoretic model of a polymorphic cancer cell population where the treatment-induced resistance is a quantitative evolving trait. When stabilization of the tumor burden is possible, we expand the model into a Stackelberg evolutionary game, where the physician is the leader and the cancer cells are followers. The physician chooses a treatment dose to maximize an objective function that is a proxy of the patient’s quality of life. In response, the cancer cells evolve a resistance level that maximizes their proliferation and survival. Assuming that cancer is in its ecological equilibrium, we compare the outcomes of three different treatment strategies: giving the maximum tolerable dose throughout, corresponding to the standard of care for most metastatic cancers, an ecologically enlightened therapy, where the physician anticipates the short-run, ecological response of cancer cells to their treatment, but not the evolution of resistance to treatment, and an evolutionarily enlightened therapy, where the physician anticipates both ecological and evolutionary consequences of the treatment. Of the three therapeutic strategies, the evolutionarily enlightened therapy leads to the highest values of the objective function, the lowest treatment dose, and the lowest treatment-induced resistance. Conversely, in our model, the maximum tolerable dose leads to the worst values of the objective function, the highest treatment dose, and the highest treatment-induced resistance.

Prostate-specific antigen (PSA) is the most commonly used serum marker for prostate cancer. It plays a role in cancer detection, treatment monitoring, and more recently, in guiding adaptive therapy protocols, where treatment is alternated based on PSA levels. However, the relationship between PSA levels and tumor volume remains poorly understood. Empirical evidence suggests that different cancer cell types produce varying amounts of PSA. Despite this, current mathematical cancer models often assume either that all cell types contribute equally to PSA levels or that only certain subpopulations produce PSA at fixed rates. In this study, we compare Zhang et al.’s classical adaptive therapy protocol with the standard of care, which involves continuous maximum tolerable dose treatment, under different assumptions regarding PSA production. Specifically, we explore the possibility that testosterone-dependent, testosterone-producing, and testosterone-independent cells contribute to PSA production to varying degrees. We use the time to competitive release as a proxy for the time to disease progression. Our findings indicate that adaptive therapy consistently results in a longer time to competitive release compared to the standard of care, regardless of the assumptions about PSA production. However, when testosterone-independent cells are the sole PSA producers, Zhang et al.’s adaptive therapy protocol becomes inapplicable, as PSA levels never fall to half of their initial value, preventing therapy discontinuation. Additionally, we observe that the number and duration of treatment cycles in adaptive therapy are highly sensitive to assumptions about how much each cell type contributes to PSA production. Overall, our results emphasize the need for a deeper understanding of patient-specific PSA dynamics, which could enhance the effectiveness of adaptive therapy in prostate cancer treatment.

Nature exhibits rapid evolution in response to human activities. When using natural resources for their own profit, humans should account for such responses. Stackelberg evolutionary games (SEG) offer a method for modeling interactions between a rational leader (humans) and evolutionary followers (nature). The followers evolve according to the principles of natural selection, and the leader tries to steer these inevitable responses in a desired direction. While the separate elements of this method, Stackelberg and evolutionary game theory, are well established, their joint realization in SEG theory is underdeveloped. Thus far, simple examples and formalisms of SEGs have considered models where the manager and evolving species have a scalar-valued controller and scalar-valued trait, respectively. Here we provide examples from cancer therapy, fisheries management, and pest control to illustrate extensions of SEG theory, where managers are attempting to control a Darwinian system. The models we develop and present highlight extensions of SEG theory to include vector-valued management strategies and vector-valued traits in the evolving species, and traits influencing different life-history stages of the species under management. Throughout we highlight the mathematical challenges that lie ahead.

Stackelberg evolutionary game (SEG) theory combines classical and evolutionary game theory to frame interactions between a rational leader and evolving followers. In some of these interactions, the leader wants to preserve the evolving system (e.g. fisheries management), while in others, they try to drive the system to extinction (e.g. pest control). Often the worst strategy for the leader is to adopt a constant aggressive strategy (e.g. overfishing in fisheries management or maximum tolerable dose in cancer treatment). Taking into account the ecological dynamics typically leads to better outcomes for the leader and corresponds to the Nash equilibria in game-theoretic terms. However, the leader's most profitable strategy is to anticipate and steer the eco-evolutionary dynamics, leading to the Stackelberg equilibrium of the game. We show how our results have the potential to help in fields where humans try to bring an evolutionary system into the desired outcome, such as, among others, fisheries management, pest management and cancer treatment. Finally, we discuss limitations and opportunities for applying SEGs to improve the management of evolving biological systems. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

We consider two-player zero-sum differential games of fixed duration, where the running payoff and the dynamics are both linear in the controls of the players. Such games have a value, which is determined by the unique viscosity solution of a Hamilton–Jacobi-type partial differential equation. Approximation schemes for computing the viscosity solution of Hamilton–Jacobi-type partial differential equations have been proposed that are valid in a more general setting, and such schemes can of course be applied to the problem at hand. However, such approximation schemes have a heavy computational burden. We introduce a discretized and probabilistic version of the differential game, which is straightforward to solve by backward induction, and prove that the solution of the discrete game converges to the viscosity solution of the partial differential equation, as the discretization becomes finer. The method removes part of the computational burden of existing approximation schemes.

The application of evolutionary and ecological principles to cancer prevention and treatment, as well as recognizing cancer as a selection force in nature, has gained impetus over the last 50 years. Following the initial theoretical approaches that combined knowledge from interdisciplinary fields, it became clear that using the eco-evolutionary framework is of key importance to understand cancer. We are now at a pivotal point where accumulating evidence starts to steer the future directions of the discipline and allows us to underpin the key challenges that remain to be addressed. Here, we aim to assess current advancements in the field and to suggest future directions for research. First, we summarize cancer research areas that, so far, have assimilated ecological and evolutionary principles into their approaches and illustrate their key importance. Then, we assembled 33 experts and identified 84 key questions, organized around nine major themes, to pave the foundations for research to come. We highlight the urgent need for broadening the portfolio of research directions to stimulate novel approaches at the interface of oncology and ecological and evolutionary sciences. We conclude that progressive and efficient cross-disciplinary collaborations that draw on the expertise of the fields of ecology, evolution and cancer are essential in order to efficiently address current and future questions about cancer.

Evolutionary game theory mathematically conceptualizes and analyzes biological interactions where one’s fitness not only depends on one’s own traits, but also on the traits of others. Typically, the individuals are not overtly rational and do not select, but rather inherit their traits. Cancer can be framed as such an evolutionary game, as it is composed of cells of heterogeneous types undergoing frequency-dependent selection. In this article, we first summarize existing works where evolutionary game theory has been employed in modeling cancer and improving its treatment. Some of these game-theoretic models suggest how one could anticipate and steer cancer’s eco-evolutionary dynamics into states more desirable for the patient via evolutionary therapies. Such therapies offer great promise for increasing patient survival and decreasing drug toxicity, as demonstrated by some recent studies and clinical trials. We discuss clinical relevance of the existing game-theoretic models of cancer and its treatment, and opportunities for future applications. Moreover, we discuss the developments in cancer biology that are needed to better utilize the full potential of game-theoretic models. Ultimately, we demonstrate that viewing tumors with evolutionary game theory has medically useful implications that can inform and create a lockstep between empirical findings and mathematical modeling. We suggest that cancer progression is an evolutionary competition between different cell types and therefore needs to be viewed as an evolutionary game.

Fish populations subject to heavy exploitation are expected to evolve over time smaller average body sizes. We introduce Stackelberg evolutionary game theory to show how fisheries management should be adjusted to mitigate the potential negative effects of such evolutionary changes. We present the game of a fisheries manager versus a fish population, where the former adjusts the harvesting rate and the net size to maximize profit, while the latter responds by evolving the size at maturation to maximize the fitness. We analyze three strategies: i) ecologically enlightened (leading to a Nash equilibrium in game-theoretic terms); ii) evolutionarily enlightened (leading to a Stackelberg equilibrium) and iii) domestication (leading to team optimum) and the corresponding outcomes for both the fisheries manager and the fish. Domestication results in the largest size for the fish and the highest profit for the manager. With the Nash approach the manager tends to adopt a high harvesting rate and a small net size that eventually leads to smaller fish. With the Stackelberg approach the manager selects a bigger net size and scales back the harvesting rate, which lead to a bigger fish size and a higher profit. Overall, our results encourage managers to take the fish evolutionary dynamics into account. Moreover, we advocate for the use of Stackelberg evolutionary game theory as a tool for providing insights into the eco-evolutionary consequences of exploiting evolving resources.

Rapid evolution is ubiquitous in nature. We briefly review some of this quite broadly, particularly in the context of response to anthropogenic disturbances. Nowhere is this more evident, replicated and accessible to study than in cancer. Curiously cancer has been late - relative to fisheries, antibiotic resistance, pest management and evolution in human dominated landscapes - in recognizing the need for evolutionarily informed management strategies. The speed of evolution matters. Here, we employ game-theoretic modeling to compare time to progression with continuous maximum tolerable dose to that of adaptive therapy where treatment is discontinued when the population of cancer cells gets below half of its initial size and re-administered when the cancer cells recover, forming cycles with and without treatment. We show that the success of adaptive therapy relative to continuous maximum tolerable dose therapy is much higher if the population of cancer cells is defined by two cell types (sensitive vs. resistant in a polymorphic population). Additionally, the relative increase in time to progression increases with the speed of evolution. These results hold with and without a cost of resistance in cancer cells. On the other hand, treatment-induced resistance can be modeled as a quantitative trait in a monomorphic population of cancer cells. In that case, when evolution is rapid, there is no advantage to adaptive therapy. Initial responses to therapy are blunted by the cancer cells evolving too quickly. Our study emphasizes how cancer provides a unique system for studying rapid evolutionary changes within tumor ecosystems in response to human interventions; and allows us to contrast and compare this system to other human managed or dominated systems in nature.

In the absence of curative therapies, treatment of metastatic castrate-resistant prostate cancer (mCRPC) using currently available drugs can be improved by integrating evolutionary principles that govern proliferation of resistant subpopulations into current treatment protocols. Here we develop what is coined as an ‘evolutionary stable therapy’, within the context of the mathematical model that has been used to inform the first adaptive therapy clinical trial of mCRPC. The objective of this therapy is to maintain a stable polymorphic tumor heterogeneity of sensitive and resistant cells to therapy in order to prolong treatment efficacy and progression free survival. Optimal control analysis shows that an increasing dose titration protocol, a very common clinical dosing process, can achieve tumor stabilization for a wide range of potential initial tumor compositions and volumes. Furthermore, larger tumor volumes may counter intuitively be more likely to be stabilized if sensitive cells dominate the tumor composition at time of initial treatment, suggesting a delay of initial treatment could prove beneficial. While it remains uncertain if metastatic disease in humans has the properties that allow it to be truly stabilized, the benefits of a dose titration protocol warrant additional pre-clinical and clinical investigations.