Trade-off between reproduction and mobility prolongs organisms' survival in rock-paper-scissors models

There is plenty of evidence that organisms' behaviour plays a central role in ecosystem formation and stability [1–5]. Because of this, one of the main goals in ecology is understanding animal adaptability to changes in environmental conditions [6]. Discovering the organisms' response to individual needs for natural resources has helped ecologists describe the relationships among spatial interactions and population dynamics [6,7]. For example, the ability to identify hostile regions or lack of natural resources allows animals to flee from enemies and find areas propitious to species proliferation [8–11]. Other animals perform self-adaptive strategies, adjusting migratory behaviour in response to internal signals, like individual energy shortage, without needing environmental cues [12–14].

It has also been shown that behavioural strategies represent an evolutionary capability which gives species an advantage in the competition in cyclic spatial games and, consequently, affects coexistence probability [15–21]. The knowledge of natural behavioural strategies has also helped the development of recent generations of robots, whose movement emulates the animals' locally adaptive methods [22]. Many inspired algorithms based on animals' self-adaptive foraging behaviour have been proposed for performance optimisation of computer systems in response to changing conditions [23].

In this letter, we investigate the spatial rock-paper-scissors model, where organisms may face energy depletion due to failed attempts to conquer natural resources by eliminating other individuals in the spatial game [24–26]. This issue has been addressed in May-Leonard models of two competing species, revealing that the system is stabilised by the significant number of deaths due to lack of energy [27]. However, researchers have not explored the effects of organisms' adaptive strategies to recover energy to prolong survival. Here, we introduce a trade-off between reproduction and mobility performed by individuals of one out of the species whenever needing energy rehabilitation [28–31]. This means weak individuals may redirect energy from reproduction to mobility, aiming to maximise the explored area, thus improving the chances of success in the spatial game [32–39].

Our model considers a three-state energy configuration —high, intermediate, and low. Accordingly, an organism has a $100\%$ chance of winning the competition in the cyclic spatial game only if it is at the high-energy level. This means that intermediate and low-energy organisms may face reversal selection interaction from an individual whose species is inferior in cyclic dominance [40]. During our research, we address the questions: i) how does the trade-off between reproduction and mobility impact the spatial organisms' organisation? ii) Is the self-adaptive trade-off effective in helping individuals recover higher energy levels, thus avoiding death by energy loss? iii) Does the behavioural strategy interfere with individuals' vulnerability to being killed in the spatial rock-paper-scissors game? iv) How does the behavioural tactic affect the organisms' survival time?

We investigate a cyclic model of three species that outcompete each other according to the rock-paper-scissors game rules, illustrated in fig. 1. The solid black arrows indicate that individuals of species i are superior in the cyclic spatial game over organisms of species i + 1, with $i=1,2,3$, and the cyclic identification $i=i+3\,\alpha$, where α is an integer. Our model posits that the organisms of species i can defeat those of species i + 1 to secure natural resources. But the organism may experience energy depletion and decreased viability if this is not achieved. When an organism of species i faces weakness due to energy loss, the cyclic advantage may be reverted. In this scenario, individuals of species i may be killed by organisms of species i − 1, as illustrated by the dashed purple arrows in fig. 1.

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Energy levels are classified into three categories: low (designated as 1), intermediate (defined as 2), and high (assigned as 3). The energy transitioning occurs every time a selection interaction is attempted: in the case of success, the energy level may either increase or maintain the highest level. Conversely, in the event of failure, the energy level may decrease, or individuals with low energy may die. The organisms' energy level defines the success in the selection interactions:

• A high-energy organism of species i always manages to eliminate an individual of species i + 1, regardless of the energy state of the defeated organism.
• An intermediate-energy organism of species i experiences reduced strength, compromising its cyclic advantage in competition against high-energy individuals. Therefore, reversal selection is possible, resulting in its elimination by a high-energy organism of species i + 1.
• For low-energy organisms of species i, the likelihood of winning against high-energy individuals of species i + 1 is further decreased. The probability of high-energy individuals of species i + 1 reversing the selection and eliminating organisms of species i is twice as likely for intermediate-energy levels.

In our model, intermediate and low-energy organisms of species 1 respond to energy depletion by performing a trade-off between reproduction and mobility. The goal is to redirect energy expenditure from producing offspring to increasing the dispersal rate, thus expanding the search region to enhance the likelihood of discovering vulnerable organisms, thus facilitating the recovery of energy levels. This strategy is not motivated by external stimuli but by the need to strengthen personal fitness, to minimise the chances of dying by energy depletion. Therefore, we introduce the trade-off factor, β, a real parameter ranging from 0 to 1, which indicates the proportion of energy redirected from reproduction to mobility in response to energy shortage.

Stochastic simulations

Let us start by observing the microscopic effects of the trade-off between reproduction and mobility on the organisms' spatial interactions. The emergence of symmetric spiral waves as spatial patterns in the standard rock-paper-scissors model (without the trade-off strategy) can be attributed to the cyclic nonhierarchical spatial game [32,42]. Therefore, to assess the influence of the trade-off strategy on the dynamics of organisms' spatial organisation, we now run a single simulation starting from the prepared initial conditions presented in fig. 2(a), where individuals are initially allocated in single-species torus rings. This means that each species initially fills one third of the grid. This approach is a commonly used technique for studying how the introduction of symmetry breaking affects spatial patterns in the standard rock-paper-scissors model [18,44,48].

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The realisation ran in a lattice with 300² grid sites for a timespan of 1000 generations and the following model parameters: $s=r=m=1/3$, $\varepsilon=0.5$, and $\beta=0.9$. figs. 2(b)–(j) show the spatial organisms' organisation at t = 25, t = 85, t = 115, t = 160, t = 200, t = 260, t = 305, t = 405, respectively. We used the colours in fig. 1 by depicting individuals of species 1, 2, and 3 with green, yellow, and pink dots; additionally, empty spaces were represented by black dots. The spatial pattern dynamics during the entire simulation is shown in the video https://youtu.be/8eaQEiKyA8M.

When the simulation commences, organisms of species i start eliminating individuals of species i − 1, thus producing the rotation of the rings from left to right. Only organisms in the ring front boundary can access individuals of dominated species; thus, the proportion of individuals failing in the selection interaction is higher far from the border. This leads to individuals' weakening and consequent deaths due to lack of energy, which creates empty spaces, as depicted by black dots inside the single-species spatial domains in fig. 2(b).

The acceleration of intermediate and low-energy individuals of species 1 results in an enlargement of the width of the pink torus ring. This effect arises from the fact that, although these individuals move faster, their movement lacks directional orientation. Consequently, the rate of individuals from species 1 entering areas occupied by species 3 increases; this facilitates the conquest of new territory by individuals of species 3.

As intermediate- and low-energy individuals of species 1 (green) perform the trade-off strategy, redirecting $90\%$ of effort from reproduction to mobility, the concentration of vacant spaces within the green area becomes higher than inside yellow and pink regions, as observed in fig. 2(b). The consequence is that many individuals of species 2 manage to infiltrate the green areas because the higher density of vacant sites represents an opportunity to reproduce within the regions of dominant species, as shown in fig. 2(c). Because newborn individuals of species 2 are in a high-energy state, they may defeat intermediate and low-energy organisms of species 1, making it possible for some of them to survive until reaching the pink area, as depicted in fig. 2(d). From this point on, the proliferation of species 2 (yellow) by eliminating individuals of species 3 (pink), produces a wave which spreads as shown in fig. 2(e)–(g). The unevenness in the spatial rock-paper-scissors game introduced by the trade-off between reproduction and mobility leads to asymmetric waves with the spatial domains of species 1 being constantly invaded by individuals of species 2, as observed in figs. 2(h)–(j).

We assume a three-state energy approach to categorise individuals based on their energy levels, namely high (level 3), intermediate (level 2), and low (level 1) energy. To recover its energy, a weakened organism must succeed in the cyclic game. Therefore, we quantify the effectiveness of the trade-off survival strategy, computing the chances of the individuals recovering when losing energy.

To explore the transitioning of the organisms' energy levels, we introduce the following recovery rates:

• $\varsigma_i^{1 \to 2}$: The likelihood of a low-energy individual of species i transitioning to an intermediate-energy state per unit time.
• $\varsigma_i^{2 \to 3}$: The probability of an intermediate-energy organism of species i changing to a high-energy state per unit time.

The implementation of these quantities follows the steps: i) counting the number of low- and intermediate-energy individuals of species i at the beginning of each generation; ii) calculating the number of organisms of species i whose energy grows during the generation, distinguishing between intermediate- and low-energy ones; iii) $\varsigma_i^{1 \to 2}$ is the ratio between the number of low-energy individuals of species i transitioning to an intermediate-energy state and the number at the beginning of each generation; iv) $\varsigma_i^{2 \to 3}$ is the ratio between the number of intermediate-energy individuals of species i going to a high-energy state and the number at the beginning of each generation.

We performed sets of 100 simulations starting from different initial conditions for $0 \leq \beta \leq 1.0$, in intervals of $\Delta \beta=0.1$. The simulations ran in lattices with 500² grid sites for a timespan of 5000 generations. To guarantee the quality of the results, we remove the data from the initial simulation stage, thus calculating the average recovery rates in the second half of each realisation. Figures 3(a) and (b) display the impact of β on $\varsigma_i^{1 \to 2}$ and $\varsigma_i^{2 \to 3}$, respectively. Green, yellow, and pink lines show the average energy recovery rates for individuals of species 1, 2, and 3, respectively, with error bars indicating the standard deviation.

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The results show the effectiveness of the trade-off between reproduction and mobility in improving the chances of energy recovery. Comparing the outcomes in figs. 3(a) and (b), one sees that the likelihood of an organism transitioning from low to intermediate energy levels is lower than the chances of transitioning from intermediate to high energy levels, regardless of the species. This happens because low-energy individuals are more susceptible to elimination through reversal selection interactions.

Furthermore, the more significant the proportion of effort redirected to increase the dispersion rate, the higher the probability of energy comeback. For $\beta=1.0$, $\varsigma_1^{1 \to2}$ is at its highest, with low-energy individuals having nearly three times the chances of recovery, compared to the case where the adaptive strategy is absent.

Regarding species 2 and 3, whose organisms do not employ the behavioural strategy, our results demonstrate both advantages and disadvantages for energy recovery that depend on the trade-off factor of species 1. For species 2, intermediate-energy individuals are helped with increased chances of recovering, reaching a peak at $\beta=0.8$; however, the prospects of transitioning of low-energy individuals worsen for any β. On the other hand, for species 3, both intermediate- and low-energy individuals are negatively affected if $\beta < 0.7$, but $\varsigma_3^{1 \to2}$ grows for $\beta \geq 0.7$, thereby favouring low-energy organisms.

We now investigate how the trade-off strategy influences the organisms' survival probability of species i. For this purpose, we first calculate the death rate, differentiating deaths by energy insufficiency from selection in the spatial game.

First, we introduce the depletion risk $\omega_i$: the probability of an organism of species i perishing due to weakness per unit time. This is implemented as follows: i) counting the number of individuals of species i at the beginning of each generation (irrespective of the energy level); ii) computing how many individuals of species i die because of energy lack during the generation; iii) computing the depletion risk as the rate of the number of dead individuals and the total number at the beginning of each generation.

Second, we compute the selection risk $\zeta_i$: the probability of an organism of species i being eliminated in the spatial rock-paper-scissors game per unit time. The implementation follows the steps: i) counting the number of individuals of species i at the beginning of each generation (irrespective of the energy level); ii) computing how many individuals of species i are killed by adversaries in the spatial game during the generation; iii) computing the selection risk as the rate of the number of killed individuals and the total number at the beginning of each generation.

We use data from collections of 100 simulations starting from different initial conditions for $0 \leq \beta \leq 1.0$, in intervals of $\Delta \beta=0.1$. The simulations ran in lattices with 500² grid sites for a timespan of 5000 generations. We removed the fluctuations in data that occur during the formation of spatial patterns in the initial stage of simulations by utilising the average depletion and selection risks found from the second half of each simulation run. Figures 4(a) and (b) depict the impact of β on $\omega_i$ and $\zeta_i$, with green, yellow, and pink lines representing species 1, 2, and 3, respectively; the error bars indicate the standard deviation.

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The outcomes unveil that the survival tactic employed by individuals of species 1 effectively protects against energy-loss–related fatalities and from enemy attacks in the cyclic game, as depicted by the green lines in figs. 4(a) and (b), respectively. As β grows, individuals of 2 and 3 also profit from the strategic behavioural strategy of individuals of species 1: the chances of death due to lack of energy decrease. However, they become more vulnerable to dying by being caught by enemies, as shown by the yellow and pink lines in fig. 4(b).

Finally, we compute the expected survival time of individuals of species i, using $\mathcal{T}_i = \int_0^{\infty}~\mathcal{S}^{t}_i\mathrm{d}t$, with the survival probability being a function of the trade-off factor given by $\mathcal{S}_i (\beta) = 1 - \omega_i (\beta) - \zeta_i (\beta)$.

Figure 5 shows the mean organisms' survival time in terms of the trade-off factor, averaged from sets of 100 simulations in lattices with 500² grid sites, running until 5000 generations; the error bars indicate the standard deviation. Green, yellow, and pink lines depict the results for species 1, 2, and 3, respectively. Accordingly, organisms of species 1 live longer as more energy is redirected from reproduction to mobility: the maximum relative growth in $\mathcal{T}_1$ is $43.74\%$, reached for $\beta=1.0$. In contrast, the expected survival time of individuals of species 3 significantly reduces, with organisms living $22.39\%$ less, on average, for $\beta=1.0$. In the case of species 2, $\mathcal{T}_2$ slightly grows for $\beta \leq 0.5$, with the relative increase reaching $0.94\%$ for $\beta = 0.3$; however, the mean survival time of individuals of species 2 drops for $\beta > 0.5$, with the maximum relative decrease, $15.75\%$, occurring when intermediate- and low-energy organisms of species 1 redirect to mobility the total energy usually spent in reproduction activity.

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Studying the stochastic version of the spatial rock-paper-scissors game, we explored the benefits of the trade-off between mobility and reproduction on organisms' energy levels and survival. For this, we consider a three-state energy approach, where individuals are weakened when they fail to take the resources from an adversary in the cyclic game. This leads to a vulnerability in playing the spatial game, thus accelerating the demise by energy depletion.

Our simulations revealed that if intermediary and low-energy organisms prioritize mobility over reproduction: i) the success in the cyclic spatial game rises, thus being more likely for an individual to regain energy; ii) the risk of being eliminated by an adversary in the cyclic game drops. Because of this, performing the trade-off between reproduction and mobility benefits organisms with prolonged mean survival time. Regarding individuals of the other species, there is a rise in the selection risk, which reduces their life expectancy.

Although we have used a specific set of interaction probabilities to run all simulations in this letter, we performed tests using other parameters. Our observation reveals that an increase in selection probability results in more failed attempts and, consequently, a more significant number of intermediate- and low-energy organisms. This provokes a frequent triggering of the mobility trade-off. On the other hand, an increase in reproduction probability facilitates energy conversion into movement, thereby maximising the benefits of the trade-off strategy.

The evolutionary trade-off strategy can significantly impact the average organisms' survival time, with benefits for one species but adverse effects for others. Therefore, this adaptive strategy, which aims to minimize the death risk, directly interferes with biodiversity stability. Further study can explore the role of organisms' conditioning in coexistence probability, focusing on the potential trade-offs between reproduction and mobility, which could promote or jeopardise biodiversity.

We thank CNPq, ECT, Fapern, and IBED for financial and technical support.

Data availability statement: The data that support the findings of this study are available upon reasonbale request from the authors.