The effect of territorial awareness in a three-species cyclic predator–prey model

Model

To investigate the evolution of cyclically competing species with intraspecific interaction which sensitively plays to the territory awareness, we employ the spatial RPS model^11,19,20,23. At the microscopic level, the model can be demonstrated on a lattice system, and for convenience, we consider a square lattice of size N with periodic boundary conditions where all sites have von Neumann neighbors. Each site can be occupied by an individual from one of the three species (referred to as A, B, and C, respectively) or left empty(E), and thus the system describes a limited carrying capacity. In addition, to explore the effect of territory awareness on intraspecific interaction, we assume that the given lattice is divided into two areas of the same size which may possibly realize different territorial ranges. Here we simply divide the two regions into the top and bottom halves of the given square lattice. To reflect the territorial awareness on intraspecific interaction, we distribute population in each group into two sub-networks randomly, and denote species (X_1) for the top and (X_2) for the bottom ((X in {A, B, C})) to distinguish the emergence of intraspecific interaction between individuals who lie on different domains. The distribution of all species with respect to the separation of the domain is illustrated in Fig. 1a.

Under the given assumption for the lattice, all the interactions between individuals occur within nearest neighboring sites by the following set of rules (see Fig. 1b):

where (i, j=1, 2). The mark (otimes) stands for any species or empty sites. Relation (1) describes interspecific interaction among three species which occurs cyclically with a rate (p_1): (A_{i}) dominates (B_{i}), (B_{i}) dominates (C_{i}), and (C_{i}) dominates (A_{i}) ((i=1, 2)). The defeated individual dies and the site becomes an empty site. Relation (2) demonstrates the intraspecific interaction which will sensitively depend on territorial awareness. Since we assume the intraspecific interaction is related to the territorial consciousness, the rate in each species may be defined by (k_{A} cdot p_{2}), (k_{B} cdot p_{2}), (k_{C} cdot p_{2}) for species A, B, C, respectively, where (p_{2}) is the given rate of interaction, k is the the sensitive parameter to territorial awareness. Similar to previous works, the result of intraspecific interaction eventually results in a death of one individual at random with a 1/2 chance. Relation (3) stands for the reproduction with a rate r which is allowed when an empty site in neighbors is selected, and migration defined by an exchange between two neighboring sites is denoted in Relation (4). Based on the theory of random walks³⁸, it occurs with a rate (m=2MN) where M and N indicate individuals’ mobility and a system size, respectively, as usual to previous works. Thus, an actual time step is defined when each individuals has interacted with others once on average, i.e., N pairwise interactions will occur in one actual time step unit. In order to make an unbiased comparison with previous works^15,19,20,21 and for the convenience of interpretations, we assume parameters as (p_{1}=p_{2}=r=1) and (k_{A}=k_{B}=k_{C}=k) (see the Methods for the meanings of specific parameters) in our simulations. Three species are divided into two types to distinguish distributions on different regions: (A_{1,2}), (B_{1,2}), and (C_{1,2}), and randomly distributed initially on a square lattice of size (N= 300 times 300). In addition, in all our simulations, species coexistence refers to the coexistence of (A_i), (B_j), and (C_k) for any combination of (i,,j,,k in left{ 1,,2 right}).

Biodiversity under territorial awareness

We first consider the effect of territorial awareness on species biodiversity. In general, it is well-known that, the spatial RPS game exhibits a transition of survival states from coexistence to extinction (which is presented by the uniform state) as individuals’ mobility increases. The phase transition occurs when M exceed a certain value, referred to as a critical mobility (M_c = (4.5 pm 0.5) times 10^{-4}), which is identified in Ref.¹¹. To address the effect of territorial awareness, we consider two different mobility values (M=1 times 10^{-5}) and (M=1 times 10^{-3}) which eventually yield different survival states: coexistence and extinction, respectively, for different sensitivity parameter k.

In general, the total simulation time T in classic spatial RPS games is considered as (T = N) which can yield the extinction for the critical mobility (M_c)¹¹. In this regard, using the time (T=N) may yield different results for species evolution and corresponding survival states due to stochastic events, and such behaviors may be induced by the choice of mobility. In our simulations, since we consider two different mobility values where the one (Fig. 2a–c) is quite lower and the other (Fig. 2d–f) is higher than (M_c), we thus consider different simulation times at (M=1 times 10^{-5}) and (M=1 times 10^{-3}): more than 490, 000 and 180, 000 steps, respectively, to obtain robust features on species survival states. The time dependent evolution of densities are illustrated in Fig. 2 where the top and bottom panels are obtained from simulations with the first 250, 000 and 140, 000 steps, respectively.

Even if different k are considered, the panels in Fig. 2 show features similar to previous works^{11,14,15,16,18,20}: coexistence and extinction for tops and bottoms, respectively. At the low mobility (M=1times 10^{-5}) as shown in Fig. 2a–c, even if the individuals located in different domains in each group disappear, the spatial RPS game eventually exhibits coexistence as k increases since some of individuals in A, B, C are survived. For instance, in our simulations, coexistence can be presented by survival of species (A_1), (B_1), (C_1) (Fig. 2a–b) or (A_2), (B_2), (C_1) (Fig. 2c). Since the typical waiting time for extinction is exponentially increasing to the size N at low mobility¹¹, there will be extinction and eventually only one species will dominate the system after extremely long times. Thus, within the finite time steps, one type of each species will disappear slowly with the increase of k and the system exhibits coexistence.

On the other hand, the high mobility (M=1times 10^{-3}) leads the extinction and only one species dominate the whole domain. As shown in Fig. 2d–f, the extinction that is defined by the two types of the species disappear occurs and the only one species finally dominate the system [e.g., (C_2), (B_2), and (C_1) for (k=5), 10, and 20, respectively]. In this case, the increase in k has little effect on the disappearance of one of the species, but has a tendency to accelerate the complete extinction of the second species. Take Fig. 2d for example, when species (A_2), (B_1) and (B_2) became extinct, (A_1), (C_1) and (C_2) is left in the system, the density of (A_1) in the system had an absolute advantage, while (C_1) and (C_2) had intraspecific interaction. Since the intensity of intraspecific interaction sensitive to territorial awareness was greater than that of interspecific interaction, the interaction was mainly intraspecific interaction between (C_1) and (C_2), then (C_1) was defeated into extinction, (C_2) preyed on the only specie (A_2) remaining in the system and eventually occupied the whole system. As k value affects the intraspecific interaction intensity, it determines the waiting time for the extinction of two species in the system. For example, we found that the larger the k value is, the shorter the waiting time for the extinction of two species is, as shown in Fig. 2e–f. But this is only the observation result of a single simulation. Due to the randomness of the simulation, this phenomenon needs further verification, so we give specific results about the effect of k value on the average extinction time in the next section. Due to stochastic events during Monte-Carlo simulations, the combination of survival species for coexistence and extinction at the final step can be different, but the such states at two mobility regimes will be still maintained. Fig. 2 may impose the follows: territorial awareness on intraspecific interaction can eventually yield similar feature to previous works in a broad aspect, but the composition of the surviving species type for each state may vary.

Average extinction time versus territorial awareness

While survival states in both cases are consistent with previous works on the effects of species migration in Fig. 2, we found an interesting feature that the evolutionary time when some type of species disappear is changed depending on k. To be concrete, at (M=1 times 10^{-5}), we found that one type of each species (A_1), (B_1), (C_1) (Fig. 2a) will eventually coexist while their companion species (A_2), (B_2), (C_2) are extinct as t exceeds (t approx 50{,}000). As k is increasing, the time point when one genus of each species disappears shows an increasing pattern as presented in Fig. 2b,c. The opposite trend can be captured at high mobility (M=1 times 10^{-3}), that is, the increase of k seems to shorten the evolution time of two species extinction in the system. Based on these observations, we may assume that the critical time for such disappearance phenomena has a certain relationship with k and the relation may differ to the choice of M.

To answer the issue, we measure the average extinction time T. In classic RPS games, traditionally, the extinction state on spatially extended systems has been identified by the uniform state that only one species dominates whole domain^{11,14,15,16,18,20}. As shown in Fig. 2a–c which ultimately describe a coexistence state in a finite time, however, any one of type in each species disappeared and the time associated with the phenomena is changed by the strength of k. In a slightly different aspect to the classic meaning of extinction, we here define the average extinction time T with respect to the regime of mobility: (a) the evolutionary time when one genus of each species disappears for low mobility and (b) the time when two of the three species disappear completely for high mobility. In this consideration, for both given cases of M in Fig. 2, the average extinction time T in each k is measured from 30 independent realizations and presented in Fig. 3.

As shown in Fig. 3a, we find clearly that the average extinction time is obviously affected by the strength of sensitivity coefficient k, especially, when the mobility is low. When species has no consciousness on territories ((k=0)), the system becomes exactly the classic RPS model¹¹ since intraspecific interaction is undefined, and the waiting time T generally tends to increase exponentially to the choice of M. However, our simulation shows the T is approximately measured at (T=110{,}000) at (k=0). Traditionally, it is well known that the average waiting time for extinction in the classic RPS game is taken (T=N) near the critical mobility regime ((M approx M_c)), and the coexistence duration is exponentially increasing as M decreases from (M_c). Within this knowledge, our simulation results may seem inconsistent with the general concept of extinction time. In our model, however, the definition of extinction is different at the low mobility regime, and the change into a single RPS system as one genus of each individual disappears may have a similar meaning to the previous definition of extinction in some sense, the above result can be said to be reasonable.

The important point is actually addressed for (k>0). In this case, species can allow intraspecific interaction and the strength of intraspecific interaction is also increasing since the territorial awareness is intensified. As a result, it is found that the average extinction time T shows a tendency to gradually increase with the increase of k at (M=1 times 10^{-5}). In addition, this trend can also be observed at (M = 1 times 10^{-3}), but it is more gradual. To investigate whether the tendency to prolong the waiting time for extinction time at low migration rates is caused by territorial awareness or the existence of intraspecific interaction, we compared traditional intraspecific interaction (i.e., intraspecific interaction among all individuals of the same species, regardless of territorial residence, which equivalent to removing the condition (ine j) from Relation (2)) with territorial-sensitive intraspecific interaction in our model, the results are shown in Fig. 3b. We found that in the presence of intraspecific interaction, the average extinction time increased with the intensity of intraspecific interaction. Specifically, the stronger the intraspecific interaction, the slower the loss of species diversity. However, compared with the traditional situation, intraspecific interaction influenced by territorial consciousness controlled the delay of extinction to a certain extent. Even if our simulations have been carried on for two specific M, it is obvious that the territorial awareness can affect the average extinction time, and we suggest that a strong sense of territoriality can also delay species extinction and lead to long-term coexistence of systems at low mobility regimes, although the introduction of territoriality leads to faster damage to species diversity than is traditionally the case, while it does not affect significantly on the extinction time and the biodiversity (which eventually appears as extinction) at high mobility regimes.

Evolution of the interface between territories

From the investigation on the average extinction time in Fig. 3, we know that the territorial awareness can affect not only species survival but also the maintenance period of survival states. Here, we may wonder why the territorial awareness can affect the waiting time to extinction. In order to investigate such an issue, we observe evolution of the spatial system, in particular invasion between species near the border on two territories, i.e., the evolution of the interface. To capture the phenomena in detail, we consider pattern formations associated with the given two mobility values at the initial state of the evolution (e.g. (t=1000)) which are represented in Fig. 4.

The top and bottom panels in Fig. 4 exhibit spatial patterns for the low and high mobility values, respectively. For (M=1 times 10^{-5}), when the value k is small such as (k=0.5) (see Fig. 4a), interspecific interaction can occur more frequently than intraspecific interaction among all pairwise reactions (1)–(4). The system can exhibit similar pattern formations to the classic RPS game¹¹. Three species, even if they are distinguished into six subgroups, are spirally entangled with clearly exhibiting spiral waves which are appeared in both two territories. Since the given lattice has periodic boundaries, species in both territories can migrate to the other region each other, but such migration is weak because the normalized probability for migration (Relation (4)) is small at the low mobility. Thus, when the system exhibits coexistence, it may be possible to predict that the top and bottom territories present dominance of species (X_1) and (X_2) ((X in {A, B, C})), respectively, while our simulations only present spatial patterns at the first 1,000 steps which may be too short to lead phase transitions.

We also find that the spiral-wave patterns are getting to fuzzy as k increases. In particular, such fuzzy patterns are conspicuous near the boundary between the two territories at the large k (see Fig. 4c,d). The increase of k directly means the intensification of intraspecific interaction, and according to the setting on the initial distribution of population, intraspecific interaction will have many chances to occur in the vicinity of the boundary than near the top and bottom periodic boundaries. Frequent intraspecific interaction can provide as many chances to allow reproduction as possible, and high intraspecific interaction rate can dominate on pairwise invasions than interspecific interaction.

In the vicinity of the border between two territories, the occurrence of intraspecific interaction is observed more prominently at (M=1 times 10^{-3}), and such features are clear as k increases. To be concrete, compared with figures among Fig. 4e–h, we found that empty sites are produced near the border and their presence is clear for high strength k such as 10 and 20 (Fig. 4g–h). In this case, the two domains appear to be more clearly divided and each domain is dominated by a single RPS system. Each single RPS system shows extinction state (only one genus survives) at high mobility, and eventually shows extinction state through interspecific or intraspecific interaction depending on the type of surviving genus. This is in good agreement with the results we obtained in Fig. 2. However, it can be seen from Fig. 3 that the time for each domain system to reach extinction at high mobility is very short compared to that at low mobility, and this has no relation with the degree of territorial awareness in interspecific interaction.

From our simulations, we find that: the relationship between territorial awareness and the average extinction time is particularly prominent at the low mobility, and the likelihood of intraspecific interaction is relatively high near territorial boundaries. Under these considerations, we may expect a new relationship between the delay of the extinction time and boundary of two territories. To uncover this veil, we try to quantify a width for occurrence of intraspecific interaction near the border between two area with respect to the territorial awareness. Specifically, we give each node on a two-dimensional grid a coordinate, defined by its row and column position. For each column (j=1,ldots ,L), calculate the interface width, defined as I:

with the quantity P(1, j) is the abscissa of the individual of species (X_{1},(Xin A,B,C)) in column j reaching the position nearest to the interface in its own region when any individual of species (X_1) does not exist in the territory of species (X_2), or furthest from the interface in its opponent region when one or more individuals of species (X_1) invade the territory of species (X_2). P(2, j) is similarly defined, corresponding to individuals of species (X_{2},(Xin A,B,C)). In order to avoid the effect of periodic boundary conditions, we only consider 100 row positions in the middle of the given lattice, i.e., (100<P(1,j),,P(2,j)le 200). The specific definition and numerical determination of P(1, j) and P(2, j) are illustrated in Fig. 5.

By averaging the interface width of each column, the interface width statistics of the whole system can be obtained:

by this definition, W can be understood as the average range at which species in two territories invade each other near the interface. At a particular time step, we can calculate the values of W for two mobility values (M=1 times 10^{-5}) and (1 times 10^{-3}). Here, we pay attention to the evolution of the initial stage, take time step (t=1000), and calculate the average value from 30 independent realizations, the results are shown in Fig. 6.

Figure 6a shows the dependence of the average interface width statistic W on the strength of sensitivity to the territory k. For (M=1 times 10^{-5}), the average interface width statistic W is decreasing as k increases, which means interactions among species on two different regions can only occur near the border of two area. In the rest of the area except near the border, species may have many chances to do reactions except intraspecific interaction since only species (X_1) or (X_2) ((X in {A,B,C})) can interact by itself in the rest area due to the rare event of migration at the low mobility regime, which can validate the long-term coexistence of species (X_1) and (X_2) ((X in {A,B,C})) in each rest area. Such a decreasing phenomenon is also obtained at the high mobility (M=1 times 10^{-3}) even if the decrease is lower than in the case of (M=1 times 10^{-5}). Thus, in the vicinity of the border between two regions, intraspecific interaction can occur frequently with high k and invasion, i.e., interspecific interaction, among different species groups is slowing down which eventually is consistent with Fig. 4. In addition, in order to illustrate the effective measurement of interface width statistics for the speed of system evolution, we describe the relationship between the average extinction time T and interface width statistics W in Fig. 6b by combining the results obtained in Fig. 3a. It can be seen from the figure that T decreases with W under both mobility conditions, but the trend of high mobility is not as obvious as that of low mobility. This declining trend illustrates the fact that at the same time step, the wider the interface between two regions, the faster the system evolves and reaches what we define as an extinction state. In short, larger values of k result in smaller statistics indicating slower invasion another territory of the species.

Source: Ecology - nature.com