DISCUSSION
We propose a new framework for analyzing the context dependence of population dynamics in a two-species system, in which we use the intrinsic growth rates of each species as a proxy for environmental suitability, and then assess how the strength of intra- and interspecific density dependence varies with environmental suitability. Our results showed that the strength of both intra- and interspecific density dependence decreased as the environmental suitability of the focal species increased, but the environmental suitability of the opposite species had little effect. Thus, in a two-species system the strengths of intra- and interspecific density dependence are primarily dependent upon the environmental suitability of the focal species.
Here, the strength of intraspecific density dependence decreased as the intrinsic growth rate of the focal species increased, which is contrary to the results of many previous studies in which the strength of intraspecific density dependence increased as the intrinsic growth rate increased (Lillegård et al. 2008; Zehnder & Hunter 2008; Pasinelli et al. 2011; Roy et al. 2016; Gamelon et al. 2019; Koetke et al. 2020). On the other hand, a few studies have shown a negative relationship between environmental suitability and the strength of density dependence (Agrawal et al. 2004; Lines et al. 2020). Thus, the strength of intraspecific density dependence has various responses to environmental changes among different species and habitats.
Previous studies that examined how interspecific density dependence varies along an environmental gradient revealed that increasing productivity (Fowler 1982; Gurevitch 1986; Wilson & Shay 1990; DiTommaso & Aarssen 1991; Wilson & Tilman 1991; Turkington et al. 1993; Belcher et al. 1995; Kadmon 1995; Twolan-Strutt & Keddy 1996; Sammul et al. 2000, 2006; Carlyle et al. 2010; LaManna et al. 2017) and abiotic environmental stress (Bertness & Ewanchuk 2002; Wood et al. 2010; Bennett et al. 2015; LaManna et al. 2016; Calizza et al. 2017; Clark et al. 2018; Wainwright et al. 2019) can produce an increase, decrease, or no change in the sign and strength of interspecific density dependence. The inconsistent results obtained from gradient-based approaches suggest that there is no obvious general pattern in interspecific density dependence along a certain environmental gradient. There are several possible explanations for the failure to find a general pattern in interspecific density dependence along environmental gradients. First, the sign and strengths of interspecific density dependence have different responses to productivity and environmental stress among species and habitats (Kawai & Tokeshi 2007; Rees 2013; Chamberlain et al. 2014). Furthermore, the strength of interspecific density dependence may vary nonlinearly across a single environmental gradient (Bimler et al. 2018). The framework proposed here could not be affected by these limitations of the gradient-based approach, because the different environmental gradients can be integrated into environmental suitability. Indeed, here we showed that for both C. dalli and G. furcata , the strength of interspecific density dependence of each species decreases when its own environmental suitability, expressed as intrinsic growth rate, increases. Thus, by combining an environmental gradient-based approach with our framework, our knowledge of the context-dependence of species interactions will be greatly improved, especially in terms of the pattern and underlying mechanisms of spatial variation in interspecific competition and facilitation.
The negative relationship between the strengths of both intra- and interspecific density dependence and intrinsic growth rate detected for both C. dalli and G. furcata can be explained by a unique role of substrate microtopography on intraspecific competition in rocky intertidal sessile organisms (Raimondi 1990; Noda et al. 1998). Both desiccation stress and predation intensity for these organisms vary at a small spatial scale depending on the surface microtopography of their substrate. Therefore, in an unsuitable locality, where intrinsic growth rates are low, recruitment of sessile organisms is restricted to a small number of suitable sites with a specific microtopographic condition (Raimondi 1990; Noda et al. 1998).
Based on the modern coexistence theory (Chesson 2000, 2003, 2018; Chesson & Kuang 2008; Barabás et al. 2018), a necessity condition for population persistence of species i in a two-species system (invasion criterion)—that is, species i has a positive population growth rate when its own density is low and the density of the competitor species j is at equilibrium (Broekman et al. 2019; Grainger et al. 2019)—can be written for a Gompertz model as:
Rinv,i = ri [1 – (rj / ri ) × (αij / αjj )] > 0, (4)
where Rinv,i ,ri ,rjjj , and αij are invasion growth rate of species i , intrinsic growth rate of species i , intrinsic growth rate of species j , the strength of interspecific density dependence for species j , and the strength of interspecific density dependence of species j on species i , respectively (see Supporting Information for a detailed derivation). Thus, the invasion growth rate of species i (Rinv,i ), for example, increases as increasing ri and decreasing (rj / ri ) × (αij / αjj ) . Consequently, a small difference in intrinsic growth rates between two species (i.e., a small fitness difference: rirj ) and a low ratio of the strength of interspecific density dependence to that of intraspecific density dependence (i.e., a large niche difference: small αijjj and small αjiii ) favor mutual invasion (Broekman et al. 2019; Grainer et al. 2019). In the case of neutral coexistence, the intrinsic growth rates of the two species must be the same (ri = rj ), and the strengths of intra- and interspecific density dependence of the two species must be the same (αii = αj = αjj = αji ).
These theoretical rules dictating the conditions necessary for stable coexistence derived from the modern coexistence theory illuminate several features of the coexistence mechanism operating in the C. dalliG. furcata system. First, within a wide range of environmental suitability, C. dalli and G. furcatacoexisted owing to niche differences between them. This is because the strength of intraspecific density dependence for opposite species is 6- to 10-fold larger than the strength of interspecific density dependence from opposite species, regardless of variation in the intrinsic growth rates of either species (Fig. 3). Second, neutral coexistence (Hubbell 2005), often said to be important to maintaining species diversity in various assemblages (Wootton 2005), did not occur in any situation. Even if both species exhibited similar intrinsic growth rates, the strength of intraspecific density dependence for opposite species is on average 10-fold larger than the strength of interspecific density dependence from opposite species (Fig. 3). Finally, if environmental suitability decreases for the focal species i but increases for the opposite species j the chance of successful invasion of the focal species will decrease, because decreasing ri and increasing αij / αjj decreased the invasion growth rate of the focal species for bothC. dalli and G. furcata (Fig. 4).
Our proposed framework has several limitations. First, this framework cannot be applied to situations in which invasion has failed in order to evaluate the causes, because it requires that both target species can be continuously observed. Second, the relationship between the strengths of intra- and interspecific density dependence and the intrinsic growth rates indicates pattern but not processes, because differences in intrinsic growth rates among localities will be multicausal phenomena, as suggested by the inconsistent relationship between strength of intraspecific density dependence and intrinsic growth rate in various studies (e.g., Agrawal et al. 2004; Lillegård et al. 2008; Zehnder & Hunter 2008; Pasinelli et al. 2011; Roy et al. 2016; Gamelon et al. 2019; Koetke et al. 2020; Lines et al. 2020). Compared with the framework proposed here, future work should focus on the integration of this framework with the environmental gradient approach, which could be helpful for interpreting the mechanism underlying the pattern between strength of intraspecific density dependence and intrinsic growth rate through decomposing the effect of each environmental component.
There is a longstanding debate among ecologists concerning how population parameters, such as intrinsic growth rate and the strength of density dependence, regulate population dynamics across a species distribution range. Many previous studies have explored the mechanisms leading to changes in population parameters and their effects on population dynamics or stable coexistence, whereas few studies have predicted the variation in population parameters. Our study demonstrates a new framework that allows the strengths of intra- and interspecific density dependence of pairs of co-occurring species to be predicted, using intrinsic growth rates of each species as a proxy for environmental suitability, in a two-species system. In the C. dalliG. furcata system, the strength of intra- and interspecific density dependence decreased as the focal species’ intrinsic growth rate increased, which is not consistent with previous results. This suggests that the strengths of density dependence have various responses to environmental changes among different species and habitats. Furthermore, combining this framework with modern coexistence theory can provide a deeper understanding of the coexistence mechanisms in a two-species system. However, our framework cannot be used to detect causes of failure of invasion or to reveal the mechanism by which the strengths of intra- and interspecific density dependence are related to intrinsic growth rates. Thus, future work could try to combine the environmental gradient-based approach with our framework to deepen our understanding of the context dependence of species interactions.