Philippa Kaur

As stated above this study aims to shed light on the deep uncertainties that are associated with the climate change phenomenon. The seasonally-averaged surface air temperature, hereafter simply referred to as temperature, was selected as the non-conditional climatic variable to be monitored within the Karkheh River Basin, Iran, during the baseline period (1975–2005). The CORDEX datasets (RCP 8.5) were employed to make climate-change projections.The first step in the proposed framework is to identify the most suitable theoretical distribution function to represent the stochastic behavior patterns of both historical and climate change data sets. Such identification considered the following theoretical distributions: normal, lognormal, exponential, Weibull, 3-parameter Weibull, extreme value, gamma, logistic, and loglogistic. It is important to note here that the primary strategy in this study is to analyze the data from a numeric standpoint without any presumption about the stochastic structure of the data44. As such, the study would opt for any distribution that is deemed fittest to describe the data. A summary of the fitted distributions to represent the prior distributions and likelihood functions is found in Tables S1 through S4 (see the Appendix). Furthermore, the climate-change period was divided into three mutually exclusive time frames which are short-term (2010–2039), mid-term (2040–2069), and long-term (2070–2099) future to gain a better understanding of the evoluton of future temperature changes.With Bayes’ theorem in mind, a Markov Chain Monte Carlo (MCMC) method was then applied to merge the prior distributions and the likelihood functions and to generate a sample set from the posterior distribution set. After a series of trials-and-errors, the sample size for the MCMC algorithm was set to be 1000 (n = 1000). These generated sample sets were then used to specify the most suitable theoretical pdfs to represent the posterior distribution functions. Figure 3, for instance, illustrates the most appropriate theoretical distribution that could represent the posterior distribution for the Seimareh sub-basin during spring under the short-term period.Figure 3The step-by-step process of computing the posterior pdf: (a) the prior distribution of Seimareh sub-basin during spring and the likelihood function of this sub-basin in the short-term future; (b) the histogram of the generated samples; and (c) the posterior distribution.Full size imageFigure 4 demonstrates the frequency with which each specific theoretical distribution functions was deemed the most suitable to characterize the prior, likelihood, and posterior distributions. Analyzing the fitted pdfs in Fig. 4 reveals an important point about the nature of RCMs’ raw projections. Specifically, the most frequently chosen distribution function for prior and posterior distributions is the 3-parameter Weibull. As for the likelihood function, however, it was the normal distribution that outperformed other available alternatives. Furthermore, the type of selected theoretical distribution for prior and posterior pdfs seems far more diverse compared to those from the likelihood functions. In fact, the likelihood functions were only limited to three types of distributions, most of which are normal distributions. Keep in mind that these functions are the most suitable pdfs that were fitted to the RCMs projected results. The cause behind this notion might be traced back to the nature of RCMs’ projections. RCMs operate at a finer horizontal resolution than GCMs, and thus they provide localized and high-resolution detailed climatic information that can be of importance for many management purposes, especially in regions with complex topography. However, the analyzed data revealed that among the distributions fitted for the likelihood function the normal distribution was found to be the best distribution to describe the data 70% of the time. This could be interpreted as signaling that employing RCMs’ raw projections, especially for regions that have considerable volatility in their climatic variables, should be used with caution, and further adjustment to the raw projected data may be required in some cases. Note that from a statistical standpoint, the normal distribution is not heavy-tailed, and as such, may not be the best way to portray this data. The fact that, in most cases, it has been selected as the best way to portray the stochastic nature of the likelihood function (i.e., RCM’s projections) means that innate characteristics of these data might prevent them to truly represent these types of variables on their own.Figure 4The frequency of using each individual theoretical pdfs as prior, likelihood function, and posterior distributions.Full size imageFigure 5 provides additional information regarding the frequency in which each individual theoretical distributions were deemed suitable to represent the posterior pdfs. While posterior distribution sets are, indeed, the most diverse in terms of the number of different types of distributions, a significant proportion of fitted pdfs (approximately 52%), however, are fitted by the 3-parameter Weibull distribution. Further information regarding the fitted distributions to represent the posterior pdfs is found in Tables S5 to S7 (Appendix).Figure 5The frequency of using different theoretical distributions as posterior pdfs.Full size imageThe computed posterior distribution functions can be interpreted as modified representations of the stochastic behavior of temperature variable concerning the short-term, mid-term, and long-term climate change projections. In that spirit, employing the confidence interval of 95%, the average temperature of the entire basin is depicted in Fig. 6 associated with historical and climate change conditions. Two sets of behavior patterns are observed. The first one is a broad trend in summer. The second pattern describes the rest of the seasons. In summer (Fig. 6b) the presence of a mild, yet, steady positive trend (upward) is detected. Here, one can expect the average temperature of the basin to increase steadily with the passage of time. As for the rest of the seasons, while it seems that the average temperature of the basin would experience a mild drop in the short-term, the temperature would begin to rise with a steady trend with time. In spring (Fig. 6a) and autumn (Fig. 6c) time series, it is projected that the expected average temperature in the basin would eventually surpass those that had been experienced in the baseline condition in the mid-term and long-term future. Concerning winter temperature it is seen in Fig. 6d that it is projected to increase over time. Yet, it has been estimated that it might not reach the observed average temperature of the basin in neither of the expressed time frames. Of course, given the upward trend in the data, this temperature would indeed be reached in a longer timeframe. It is worth noting that these patterns are in line with the idea that the earlier impacts of climate change are to amplify the historical patterns in climatic variables. That is why the data show a slight drop in colder seasons and an uptick in the warmer ones. That is, of course, until eventually, a new climatic equilibrium is reached on a global scale. At this point, the temperature as shown here would start to increase gradually.Figure 6The historical and simulated average temperatures of the entire basin with the 95%confidence interval in (a) spring; (b) summer; (c) autumn; and (d) winter.Full size imageThe other notable implication that can be understood by analyzing Fig. 6 is the variation in the width of the confidence intervals under baseline and climate change conditions. In comparison to the baseline condition, the length of the 95% confidence intervals would dramatically decrease under climate change conditions. This shrinking indicates that the generated results are more densely surrounding the central tendency measure herein chosen as the mean (μ) of the data. This notion is still in line with the idea that RCM’s projections mostly resemble the stochastic characteristic of normal distributions. A normal distribution is by nature not a heavily-tailed distribution, meaning that it rarely generates tail values. Even though the MCMC framework has mitigated this effect to some extent, they inevitably inherit this stochastic property from the likelihood functions.Again, to truly understand the obtained results here, one must first acknowledge how Bayesian models work. The main idea behind a Bayesian-based framework is to adjust the prior assumptions about a stochastic phenomenon through observed samples. In this case, the prior information represents the historical data, and the likelihood function (i.e., the samples) is obtained from RCM projects. As can be seen here, while RCMs’ projections might be perfectly capable of portraying the normal behavior of a variable under climate change conditions, which is usually sufficient for most lumped evaluation of climate change impact assessments, they might not be suitable to study extreme hydro-climatic events. The main problem with the raw RCM projections is that they follow a normal distribution, which is a symmetric distribution. Figure 4 suggests that while the MCMC framework here is mitigating this impact the final projections inherit this property from the likelihood functions. This simply means that while any RCM-based projection is perfectly suitable to understand the general outline of the climate change impacts, they are not the best option to study extreme events because even by modifying their pdfs, they rarely generate truly extreme values. The average temperatures in all sub-basins under baseline and climate change conditions are summarized in Table 1.Table 1 The average surface air temperature in all sub-basins under baseline and climate change conditions (°C).Full size tableAs for the impact of climate change, it is clear that these data are associated with deep uncertainty; that is, the parameters used to describe the stochastic behavior of a variable may be subjected to some degree of uncertainty. These parameters, μ for one, may also be represented by a pdf of their own. This study focuses on highlighting this type of deep uncertainty that might interfere with the central tendency measure μ.The deep uncertainty in this instance dictates that the recorded parameters for each posterior distribution are not deterministic values. While for a given prior distribution and likelihood function the MCMC would lead to a specific type of posterior pdf, the parameters that are used to define this pdf (e.g., μ), could vary each time the algorithm is used. If this variation is mild, there is more certainty about the nature of the variable’s stochastic behavior pattern (i.e., the posterior distribution function). If it is determined that the parameters are experiencing severe variations then the deep uncertain environment would leave the decision-makers unsure about the variables’ stochastic behavior pattern.With that idea in mind the combination of prior distributions and likelihood functions was executed for 100 times, and in each iteration the mean of 1000 samples was recorded. A theoretical distribution function was then fitted to the recorded values. Naturally, if the recorded values are generally close to one another numerically, the parameters of the computed posterior pdfs are less subjected to deep uncertainty. If, however, these values show significant fluctuation then the deep uncertainty of climate change would impede predictions of the stochastic behavior pattern of temperature. Figure 7, for example, portrays the uncertainty of the computed μ parameter for Seimareh sub-basin in spring under short-term future condition.Figure 7The uncertainty of the computed μ parameter for Seimareh sub-basin in spring under short-term future condition demonstrated by (a) a histogram and (b) a probability distribution function.Full size imageFigure 8 demonstrates the number of times each theoretical distribution was chosen to portray the stochastic behavior of the μ parameter. As can be seen here, the normal and lognormal distributions are the most common pdfs used to describe the variation in the μ parameter. One should also note the fact that about 65% of the distributions used to describe the future condition are normal distributions. The list of fitted pdfs is summarized in Tables S8 to S10.Figure 8The frequency with which each theoretical distribution was found suitable to describe the stochastic distribution of the μ parameter.Full size imageTable 2 summarizes the variation in the computed μ parameter in each given sub-basin. It is seen in Table 2 the 95% confidence interval of the μ parameter in all cases ranges between ± 0.1 and ± 0.3 °C. In 55% of the cases, this interval was found not to be more than ± 0.1 °C, and, furthermore, in 97% of them the interval was less than ± 0.2 °C. Needless to say, a widened confidence interval for the μ parameter can only signal that the deep uncertainty has a more pronounced impact on the temperature’s stochastic behavior. As for the case of the spring data set of the Seimareh sub-basin under the short-term condition, or the case of the Gharesou sub-basin’s winter data series under short-term period, the confidence interval for the μ parameter is estimated to be ± 0.3 °C wide. This indicates that compared to other projected posterior pdfs there is less certainty about the predicted stochastic behavior pattern of temperature variable for these particular cases. As shown in Table 2 in some cases, the variation in the projected μ temperature’ posterior pdfs is decreasing over time (for a given season over different timeframes). As discussed earlier, this was interpreted as the deep uncertainties of the climate change projections, meaning that lower volatility in this measure indicates that the said variable is less affected by the deep uncertainty of the climate-change phenomenon. This observation is in line with the general belief that, in the near future, the climate change phenomenon is most likely to intensify the historical patterns in climatic variable, but gradually we expect to see an upward trend in temperature in the longer run45. In this case, there is more volatility in the earlier time frames, but as time progresses, this volatility seems to decrease in some cases. This means that the obtained projections are showing less uncertainty about the outline behavior of the parameter for the long-term future as the models that are simulating the climatic behavior under climate change conditions have already reached a new equilibrium by that point.Table 2 The variation in the computed μ of the temperature’ posterior pdfs.Full size table More

ModelTo investigate the evolution of cyclically competing species with intraspecific interaction which sensitively plays to the territory awareness, we employ the spatial RPS model11,19,20,23. 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.Figure 1Schematic diagrams of network structure and the invasion rules among species. (a) Each circle represents a node, and individuals of species A, B, and C are evenly and randomly distributed on each node. To realize territorial awareness, the lattice is divided into two regions of equal size: the top and the bottom where the dashed line indicates the regional boundary. Two genera of the same species are distributed in different regions, and different color markers represent different species types. Nodes without color markers are empty nodes. (b) Interspecific interaction among three species A, B, and C (indicated by three boxes) occurs cyclically with a rate (p_1). A box of each group describes the intraspecific interaction between individuals who belong to different territories where the interaction is regulated by territorial consciousness. Here intraspecific interaction in each group occurs with a rate (k_i cdot p_2) ((i in {A, B, C})).Full size imageUnder 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):$$begin{aligned}&A_{i}+B_{j}overset{p_{1}}{longrightarrow }A_{i}+E,quad B_{i}+C_{j}overset{p_{1}}{longrightarrow }B_{i}+E,quad C_{i}+A_{j}overset{p_{1}}{longrightarrow }C_{i}+E, end{aligned}$$
(1)
$$begin{aligned}&A_{i}+A_{j}overset{k_{A} cdot p_{2}}{longrightarrow }A_{i}+E,mathrm{or},E+A_{j}, quad B_{i}+B_{j}overset{k_{B} cdot p_{2}}{longrightarrow }B_{i}+E,mathrm{or},E+B_{j}, quad C_{i}+C_{j}overset{k_{C} cdot p_{2}}{longrightarrow }C_{i}+E,mathrm{or},E+C_{j},quad ine j, end{aligned}$$
(2)
$$begin{aligned}&A_{i}+Eoverset{r}{longrightarrow }A_{i}+A_{i},quad B_{i}+Eoverset{r}{longrightarrow }B_{i}+B_{i},quad C_{i}+Eoverset{r}{longrightarrow }C_{i}+C_{i}, end{aligned}$$
(3)
$$begin{aligned}&A_{i}+otimes overset{m}{longrightarrow }otimes +A_{i},quad B_{i}+otimes overset{m}{longrightarrow }otimes +B_{i},quad C_{i}+otimes overset{m}{longrightarrow }otimes +C_{i}, end{aligned}$$
(4)
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 works15,19,20,21 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 awarenessWe 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.Figure 2Time dependent evolution of densities in the system for different M and k. Top and bottom panels are obtained with (M=1times 10^{-5}) and (M=1times 10^{-3}), respectively, and the sensitivity parameter k is given by (k=5), 10, and 20 from the left to right in each row. (a–c) Regardless of the choice k, the low mobility still leads species coexistence as usual. (d–f) At high mobility regimes, the system also always exhibit the extinction state.Full size imageEven if different k are considered, the panels in Fig. 2 show features similar to previous works11,14,15,16,18,20: 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 awarenessWhile 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 domain11,14,15,16,18,20. 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.Figure 3The average extinction time T as a function of the territorial sensitive parameter k. (a) Two cases of fixed mobility in territorial sensitive intraspecific interaction. For low mobility (M=1times 10^{-5}), the time T which is measured by detecting the time when one genus of each species disappears tends to increase with the increase of k, i.e., the high sensitivity of territorial awareness has the effect of delaying the waiting time for extinction. Similarly, it can be seen that at high mobility (M=1times 10^{-3}), an increase in k value will also delay the waiting time for extinction, but the effect is much more gentle. (b) At low mobility value (M=1times 10^{-5}), traditional intraspecific interaction (i.e., intraspecific interaction among all individuals of the same species, regardless of territorial residence) was compared with territorial sensitive intraspecific interaction. Here, for the traditional case, k represents intraspecific interaction intensity, and the running time of the simulation is 810, 000 steps. In the case that the final steady state has not occurred before the end of the simulation, we take the maximum time step ((t=810{,}000)) as the extinction time T value, which causes the blue line to become gentle when (k >14). Compared with the traditional situation, the intraspecific interaction affected by territorial awareness significantly reduced the average extinction time, that is, accelerated the damage of species diversity in the system. The results were averaged from 30 independent simulations, and error bars (using standard errors, which defined as the sample standard deviation divided by the square root of the number of samples) are shown in the figure.Full size imageAs 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 territoriesFrom 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.Figure 4The typical snapshots of evolution on patterns at (t=1000) for different k: 0.5 for (a) and (e), 2 for (b) and (f), 10 for (c) and (g), and 20 for (d) and (h), where the mobility is considered as (M = 1times 10^{-5}) for tops and (M=1 times 10^{-3}) for bottoms. Different colors correspond to different species types, as shown in Figs. 1 and 2, with white indicating vacancy. As k increases, the invasion among species between two territories occurs more gradually, and such phenomena are clearly observed for the high mobility as shown in the panel (h).Full size imageThe 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:$$begin{aligned} I_{j}= {left{ begin{array}{ll} P(1,j)-P(2,j),&{}quad text {if } P(1,j) >P(2,j)\ 0,&{}quad text {if } P(1,j) More

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