Introduction

Habitat selection is an important behavior that affects individual fitness, population dynamics, and species interactions (Chase 1999; Morris 2003; Takimoto 2003). For anuran adults, spawning habitat selection can critically affect their fitness, because anurans provide little parental care and juveniles stay in their natal habitats before metamorphosis (Buxton and Sperry 2017). The growth and survival of tadpoles depend on multiple abiotic and biotic characteristics of their natal habitats, such as the temperature and volume of water pools, the quality and quantity of food resources, and the abundances of competitors and predators (Herreid and Kinney 1967; Wells 1977; Resetarits and Wilbur 1989; Crump 1991; Laurila and Aho 1997). These factors interactively affect juvenile growth and survival. For example, a higher desiccation probability of temporary ponds does not only reduce the probabilities of successful metamorphosis but can also intensify intraspecific resource competition and cannibalism because other competitive and predatory species, which avoid drying ponds, are scarce (Schneider and Frost 1996; Skelly 1997; Spieler and Linsenmair 1997).

Spawning adults are thus expected to evolve abilities to discriminate the quality of habitats that their offspring will experience (Rudolf and Rödel 2005). Field and experimental evidence shows that breeding anurans assess the quality of habitats before choosing their oviposition habitats (Buxton and Sperry 2017). Because of the interacting effects of multiple biotic and abiotic factors on juvenile growth and survival, habitat selection of spawning adults can be context dependent. For example, while spawning adults may tend to avoid habitats containing many conspecific juveniles to reduce impacts of intraspecific competition and/or cannibalism on their own offspring (Crump 1991; Lin et al. 2008), adults can rather prefer to oviposit in ponds with conspecifics when conspecific presence indicates high-quality, predator-free habitats (Rudolf and Rödel 2005). In cannibalistic anurans, the timing of spawning can affect oviposition site selection. Since late-hatching juveniles are more likely to be preyed upon by early-hatching juveniles (Petranka and Thomas 1995), late-spawning adults tend to avoid habitats containing elder juveniles (Spieler and Linsenmair 1997; Iwai et al. 2007).

Despite a rich body of empirical examples, there are few formal theoretical works to analyze the context dependence of spawning habitat selection by anurans. Here we develop a mathematical model to examine the interacting effects of multiple biotic and abiotic factors on anuran spawning habitat selection. Specifically, our model will investigate how the interactions of oviposition timing, conspecific negative interactions (competition and cannibalism), and other habitat characteristics (e.g., juvenile resource abundances, heterospecific predator abundances, and other factors controlling habitat suitability for juvenile growth and survival) affect spawning habitat selection. The model will show that while early spawning adults favor habitats of superior quality, late spawning adults can prefer inferior habitats if negative effects from conspecific juveniles are large. In the discussion section, we examine conditions leading to such habitat selection and relevant empirical examples.

Model


We employ the following assumptions to formulate the model. The model considers anuran species that have a prolonged spawning season. Spawning adults deposit eggs in discrete spawning habitats that scatter across a landscape. The spawning season is divided into early and late phases. We consider that stochastic environmental conditions experienced by individual adults affect their growth and physiology that determine their spawning timing (Sheridan et al. 2018; Takahashi and Sato 2020). To model this situation, we assign the early or late phase to a spawning adult randomly according to a fixed probability. The growth and survival of juveniles depend on multiple biotic and abiotic characteristics of spawning habitats. These factors include conspecific juvenile densities, juvenile resource abundances, heterospecific predator abundances, and the physical and chemical attributes of habitats such as the volume, temperature, pH, and pollutant concentrations of habitat waters. To enhance the analytical tractability of the interacting effects of these factors, our model assumes a single composite parameter controlling habitat quality; this controlling parameter (denoted below as \(u\)) integrates the effects of multiple biotic and abiotic factors except conspecific juvenile densities. For example, the quality of habitats is low if the habitats contain abundant heterospecific predators. Temporary ponds with high desiccation probabilities represent low quality habitats when desiccation decreases juvenile survival. However, if high desiccation probabilities keep away potential competitors and predators, temporary ponds can represent high quality habitats (Schneider and Frost 1996; Skelly 1997). Based on the habitat-quality controlling parameter, we categorize spawning habitats into type-I and type-II habitats. The quality of type-II habitats is lower than that of type-I habitats. Availability of spawning habitats does not change through the entire spawning season, and juveniles do not move between habitats. In addition to habitat quality, the densities of conspecific juveniles affect their own growth and survival. Conspecific juveniles negatively affect the growth and survival of other juveniles in the same ponds through competition and cannibalism. A set of two probabilities, (\({p}_{\text{E}}\), \({p}_{\text{L}}\)), defines the habitat selection strategy of an adult. An adult breeding in the early phase selects a type-I habitat with a probability \({p}_{\text{E}}\) or a type-II habitat with a probability \(1-{p}_{\text{E}}\). Similarly, a late breeding adult selects a type-I or type-II habitat with a probability \({p}_{\text{L}}\) or \(1-{p}_{\text{L}}\), respectively. For adults ovipositing during the early phase, \({W}_{\text{I}}^{\text{E}}\) denotes the relative fitness of those ovipositing in type-I habitats, and \({W}_{\text{II}}^{\text{E}}\) the relative fitness of those ovipositing in type-II habitats. Similarly, the relative fitness of adults spawning in type-I and II habitats in the late phase are \({W}_{\text{I}}^{\text{L}}\) and \({W}_{\text{II}}^{\text{L}}\), respectively. Putting into a mathematical term, the fitness of an adult spawning in a type-I habitat in the early phase is expressed as:

$$W_{{\text{I}}}^{{\text{E}}} = \frac{{w_{0} }}{{1 + aRNhp_{{\text{E}}} /\left( {k_{{\text{I}}} K} \right)}},$$
(1a)

where \({w}_{0}\) is the baseline fitness in the absence of the effects of conspecific juveniles. The fitness is reduced as the local conspecific density, \(RNh{p}_{\text{E}}/\left({k}_{\text{I}}K\right)\), increases with a coefficient \(a\), where \(R\) is the per capita offspring production rate; \(N\) is the total abundance of spawning adults; \(h\) is the probability of an adult spawning in the early phase; \(K\) is the total number of spawning habitats; \({k}_{\text{I}}\) is the proportion of type-I habitats; and \(a\) is the strength of intraspecific negative effects among juveniles oviposited in the same spawning phase. The fitness of an adult spawning in a type-II habitat in the early phase is similarly expressed as:

$$W_{{{\text{II}}}}^{{\text{E}}} = \frac{{w_{0} \left( {1 - u} \right)}}{{1 + aRNh\left( {1 - p_{{\text{E}}} } \right)/\left( {k_{{{\text{II}}}} K} \right)}}$$
(1b)

where the habitat-quality controlling parameter, \(u\), represents the reduction of baseline fitness in type-II habitats as compared to type-I habitats; and \({k}_{\text{II}}\) is the proportion of type-II habitats (\({k}_{\text{I}}+{k}_{\text{II}}=1\)).


Spawning in the late phase can affect the fitness of spawning adults in at least three respects. First, juveniles produced in the late phase receive negative effects from conspecific juveniles produced in the early phase in addition to those among themselves. Possibly, the effects from early-hatching juveniles are larger than those from late-hatching juveniles, because early juveniles reach higher developmental stages and achieve larger body sizes, which confer an advantage over late juveniles in intraspecific competition and cannibalism. Second, in addition to increased negative effects from conspecifics, heterospecific competitors and predators that have arrived earlier can impose severe negative effects on late-hatching juveniles, because they may have priority in resource consumption and size advantages in predation (Rudolf 2022). Third, early and late-hatching juveniles may experience different abiotic conditions in their habitats. If late-hatching juveniles occur in temporary ponds, they may have severe time constraints on maturation before ponds dry up (Skelly 1997). Alternatively, small early temporary ponds can dry up without subsequent rain events (Gould et al. 2022) or become frozen due to sudden cold weather (Sjogren-Gulve and Berg 1999), risking the survival and growth of early hatching juveniles. Taking these into account, the fitness of an adult spawning in a type-I habitat in the late phase is formulated as:

$$W_{{\text{I}}}^{{\text{L}}} = \frac{{\beta w_{0} }}{{1 + aRN\left( {1 - h} \right)p_{{\text{L}}} /\left( {k_{{\text{I}}} K} \right) + \alpha RNhp_{{\text{E}}} /\left( {k_{{\text{I}}} K} \right)}},$$
(1c)

where \(\beta\) is a coefficient representing the change of baseline fitness in the late phase (due to changes in heterospecific interactions and/or abiotic conditions; \(\beta >1\) means an increase of baseline fitness and \(\beta <1\) a reduction); and \(\alpha\) is the strength of intraspecific negative effects from juveniles produced in the early phase. The magnitude of \(\alpha\) reflects the effects of cannibalism by early-hatching juveniles preying on late-hatching juveniles in addition to the effects of intraspecific competition. Finally, the fitness of an adult spawning in a type-II habitat in the late phase is similarly formulated as:

$$W_{{{\text{II}}}}^{{\text{L}}} = \frac{{\beta w_{0} \left( {1 - u} \right)}}{{1 + aRN\left( {1 - h} \right)\left( {1 - p_{{\text{L}}} } \right)/\left( {k_{{{\text{II}}}} K} \right) + \alpha RNh\left( {1 - p_{{\text{E}}} } \right)/\left( {k_{{{\text{II}}}} K} \right)}}.$$
(1d)

Analysis

We determine the evolutionary stable spawning habitat selection strategy that can resist invasion by other strategies. To do this, we employ evolutionary invasion analysis. The relative fitness of a mutant employing a strategy (\({p}_{\text{E}}{\prime}\), \({p}_{\text{L}}{\prime}\)) in the group of residents employing a strategy (\({p}_{\text{E}}\), \({p}_{\text{L}}\)) is expressed as.

$$W\left( {p^{\prime}_{E} ,p^{\prime}_{L} {|}p_{{\text{E}}} ,p_{{\text{L}}} } \right) = h\left\{ {p^{\prime}_{E} W_{{\text{I}}}^{{\text{E}}} + \left( {1 - p^{\prime}_{E} } \right)W_{{{\text{II}}}}^{{\text{E}}} } \right\} + \left( {1 - h} \right)\left\{ {p^{\prime}_{L} W_{{\text{I}}}^{{\text{L}}} + \left( {1 - p^{\prime}_{L} } \right)W_{{{\text{II}}}}^{{\text{L}}} } \right\}.$$
(2)

To quantify the potential of mutant invasion, the fitness gradients of mutant probabilities, \(p^{\prime}_{E}\) and \(p^{\prime}_{L}\), are evaluated at the resident strategy \(\left({p}_{\text{E}},{p}_{\text{L}}\right)\):

$$\left. {\frac{\partial W}{{\partial p^{\prime}_{E} }}} \right|_{{\left( {p^{\prime}_{E} ,p^{\prime}_{L} } \right) = \left( {p_{{\text{E}}} ,p_{{\text{L}}} } \right)}} = h\left( {W_{{\text{I}}}^{{\text{E}}} - W_{{{\text{II}}}}^{{\text{E}}} } \right),$$
(3a)
$$\left. {\frac{\partial W}{{\partial p^{\prime}_{L} }}} \right|_{{\left( {p^{\prime}_{E} ,p^{\prime}_{L} } \right) = \left( {p_{{\text{E}}} ,p_{{\text{L}}} } \right)}} = \left( {1 - h} \right)\left( {W_{{\text{I}}}^{{\text{L}}} - W_{{{\text{II}}}}^{{\text{L}}} } \right).$$
(3b)

These fitness gradients define the evolutionary dynamics of strategy probabilities (McGill and Brown 2007). We assume that there is no genetic correlation between the strategy probabilities, \({p}_{\text{E}}\) and \({p}_{\text{L}}\). Although such a correlation might arise if adults having a certain genetic trait tend to choose high-quality habitats irrespective of spawning phases, we do not consider such possibilities further. From the fitness gradients, we calculate the evolutionary stable strategy (ESS) of resident strategies, \(\left({p}_{\text{E}}^{*},{p}_{L}^{*}\right)\) (see supplementary information for details).

Results

ESSs of early and late spawning adults diverge as the quality of type-I and type-II habitats diverges. While early spawning adults are always more likely to choose type-I habitats of better quality (Fig. 1a–d), late spawning adults rather prefer type-II habitats of slightly or moderately poorer quality (small \(u\)) if early hatching juveniles have large negative effects on late hatching juveniles (large \(\alpha\)) (Figs. 1a, b, 2). If negative effects among juveniles of the same developmental age are sufficiently large (large \(a\)), late spawning adults prefer type-I habitats irrespective of the relative quality of type-II habitats (Figs. 1c, d, 2). Below we illustrate detailed conditions leading to these distinct strategies of early and late spawning adults.

Fig. 1
figure 1

Habitat selection by early and late spawning adults along the gradient of type-II habitat quality (\(u\)). Panels (a-d) illustrate the different habitat selection patterns resulting from various combinations of the three parameters (\(a\), \(\alpha\), \(h\)). While early spawning adults always prefer type-I habitats (\({p}_{\text{E}}^{*}\), solid gray lines), late spawning adults show qualitatively different preferences (\({p}_{\text{L}}^{*}\), black dashed lines) depending on the strength of negative effects between juveniles of the same developmental age (\(a\)) and the strength of negative effects of early hatching on late-hatching juveniles (\(\alpha\)) as well as other parameters (see Fig. 2). Larger \(u\) means lower relative quality of type-II habitats. In a (\(a\)=0.2, \(\alpha\)=0.8, \(h\)=0.5), late spawning adults increase preference for type-II habitats as \(u\) increases from zero. The preference becomes so strong that all late-spawning adults choose type-II habitats (\({p}_{\text{L}}^{*}=0\)) at intermediate \(u\). Further increase in \(u\) then increases \({p}_{\text{L}}^{*}\) and all late-spawning adults eventually choose type-I habitats (\({p}_{\text{L}}^{*}=1\)) at sufficiently large \(u\). In b (\(a\)=0.5, \(\alpha\)=0.8, \(h\)=0.5), late spawning adults prefer type-II habitats as \(u\) increases from zero, but this trend is reversed at intermediate \(u\) with \({p}_{\text{L}}^{*}\) reaching 1 at sufficiently large \(u\). In c (\(a\)=0.8, \(\alpha\)=0.5, \(h\)=0.5), late spawning adults prefer type-I over type-II habitats but less strongly than early spawning adults. In d (\(a\)=0.8, \(\alpha\)=0.5, \(h\)=0.8), late spawning adults prefer type-I over type-II habitats but more strongly than early spawning adults. Other parameters are fixed at \(RN/K\)=1 and \({k}_{\text{I}}\)=0.5. See supplementary information for the critical \(u\) values (\({u}_{\text{E}}\), \({u}_{\text{L}}^{0}\), \({u}_{\text{L}}^{1}\), \({u}_{\text{L}}^{2}\), and \({u}_{\text{L}}^{3}\))

Full size image
Fig. 2
figure 2

Effects of negative interaction strengths on the habitat preference of late spawning adults along the gradients of juvenile densities (\(RN/K\); a–d), of the probability of spawning in the early phase (\(h\); e–h), and of the proportion of type-I habitats (\({k}_{\text{I}}\); i–l). Above the solid line (\(\alpha ={\alpha }_{\text{crit}}\)) in each panel, late spawning adults prefer type-II over type-I habitats unless type-II habitat quality is very poor (Fig. 1a, b). In the area above the dashed line (\(\alpha ={\alpha }_{\text{crit}}{\prime}\)), the complete avoidance of type-I habitats by late spawning adults occurs (Fig. 1a). Between the solid and dashed lines, late spawning adults show the partial avoidance of type-I habitats (Fig. 1b). Below the solid line, late spawning adults prefer type-I over type-II habitats unless type-II habitat quality is extremely poor (Fig. 1c, d). The preference of late spawning adults for type-I habitats is weaker than that of early spawning adults (Fig. 1c) below the solid line and above the dotted line (\(\alpha ={\alpha }_{\text{crit}}^{{\prime}{\prime}}\), shown in g and h). Below the dotted line (g and h), the preference of late spawning adults for type-I habitats is stronger than that of early spawning adults (Fig. 1d). See supplementary information for \({\alpha }_{\text{crit}}{\prime}\) and \({\alpha }_{\text{crit}}^{{\prime}{\prime}}\). Values of changed parameters are \(RN/K\)= 0.05 (a), 0.5 (b), 5 (c), and 50 (d); \(h\)= 0.2 (e), 0.4 (f), 0.6 (g), and 0.8 (h); \({k}_{\text{I}}\)= 0.2 (i), 0.4 (j), 0.6 (k), and 0.8 (l). Unchanged parameters are set at \(RN/K\)= 1, \(h\)= 0.5, and \({k}_{\text{I}}\)= 0.5

Full size image

Early spawning adults generally show a stronger preference for type-I habitats as the relative quality of type-II habitat worsens (larger \(u\)). Their ESS probability of choosing type-I habitats is:

$$p_{{\text{E}}}^{*} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\tilde{p}_{{\text{E}}} } & {\left( {0 \le u < u_{{\text{E}}} } \right)} \\ \end{array} } \\ {\begin{array}{*{20}c} 1 & {\left( {u_{{\text{E}}} \le u \le 1} \right)} \\ \end{array} } \\ \end{array} } \right.,$$
(4)

where

$$\tilde{p}_{{\text{E}}} = \frac{{k_{{\text{I}}} }}{{k_{{\text{I}}} + \left( {1 - u} \right)k_{{{\text{II}}}} }}\left( {1 + \frac{{uk_{{{\text{II}}}} }}{aRNh/K}} \right),$$
(5a)
$$u_{{\text{E}}} = \frac{aRNh/K}{{k_{{\text{I}}} + \left( {aRNh/K} \right)}}.$$
(5b)

The ESS probability, \({p}_{\text{E}}^{*}\), becomes identical to the proportion of type-I habitats (\({k}_{\text{I}}\)) when type-I and type-II habitats are of the same quality (\(u\) = 0). As the relative quality of type-II habitats decreases (larger \(u\)), \({p}_{\text{E}}^{*}\) steadily increases toward one. The increase in the term \(aRNh/K\) in \({\widetilde{p}}_{\text{E}}\) makes \({\widetilde{p}}_{\text{E}}\) approach \({k}_{\text{I}}/\left\{{k}_{\text{I}}+\left(1-u\right){k}_{\text{II}}\right\}\). This indicates that the average strength of negative interactions among early hatching juveniles (determined by interaction strength [\(a\)] and the average density across all habitats [\(RNh/K\)]) weakens the advantage of early spawning adults selecting type-I habitats. At sufficiently large \(u\) (i.e., \(u\ge {u}_{\text{E}}\)), \({p}_{\text{E}}^{*}\) reaches one, meaning that all early spawning adults choose type-I habitats.

Late spawning adults show more complex habitat selection because their strategies depend on the habitat selection of early spawning adults (Fig. 1a–d). When the quality of type-II habitats is equivalent to, or slightly lower than, that of type-I habitats (i.e., small \(u\)), the ESS probability of late spawning adults choosing type-I habitats (\({p}_{\text{L}}^{*}\)) is:

$$p_{{\text{L}}}^{*} = \tilde{p}_{{\text{L}}} \equiv \frac{{k_{{\text{I}}} }}{{k_{{\text{I}}} + \left( {1 - u} \right)k_{{{\text{II}}}} }}\left\{ {1 + \frac{{uk_{{{\text{II}}}} + \alpha RNh/K}}{{aRN\left( {1 - h} \right)/K}}} \right\} - \frac{\alpha h}{{a\left( {1 - h} \right)}}p_{{\text{E}}}^{*} .$$
(6)

This expression shows that an increase in \({p}_{\text{E}}^{*}\) lowers \({p}_{\text{L}}^{*}\). In other words, early spawning adults preferring type-I habitats prevent late spawning adults from choosing these habitats. As a result, late spawning adults may not necessarily increase their preference toward type-I habitats when the relative quality of type-II habitats decreases.

To be more specific, a decrease in the relative quality of type-II habitats (i.e., the increase of \(u\) from zero) causes a reduction in the preference of late spawning adults to type-I habitats (Fig. 1a, b), if \(\alpha\) is lower than a critical value, \({\alpha }_{\text{crit}}\):

$$\alpha_{{{\text{crit}}}} \equiv a\left\{ {1 + \frac{{aRN\left( {1 - h} \right)}}{K}} \right\}.$$
(7)

This critical value is larger than \(a\), revealing a necessary condition for late spawning adults to prefer type-II habitats. That is, the strength of the negative effects of early-hatching juveniles on late-hatching juveniles must be larger than that among juveniles of the same developmental age (i.e., \(\alpha >a\)). Moreover, the critical value itself decreases as \(a\) becomes small (Fig. 2). These are an indirect consequence of smaller \(a\) making early spawning adults increase their preference for type-I habitats (Eq. 5a), which increases early hatching juveniles in type-I habitats, in turn preventing late spawning adults from choosing these habitats.

Lower densities of late-hatching juveniles decrease the critical value, \({\alpha }_{\text{crit}}\) (either through reduced \(RN/K\) [Fig. 2a–d] or reduced \(1-h\) [Fig. 2e–h]), facilitating that late spawning adults choose type-II habitats. These results show that weakening the gross impact of negative interactions among late-hatching juveniles on themselves (i.e., smaller \(aRN(1-h)/K\)) tends to make late-spawning adults choose type-II habitats. On the other hand, changing the relative proportions of type-I and II habitats does not alter the critical value (Fig. 2i–l).

Under the condition that late spawning adults prefer type-II habitats, they may completely (Fig. 1a) or partially (Fig. 1b) avoid type-I habitats at intermediate levels of \(u\) (supplementary information). The complete avoidance will occur if

$$\alpha > \alpha^{\prime}_{{{\text{crit}}}} \equiv a\left( {1 + \frac{{aRN\left( {1 - h} \right)/K}}{{k_{{{\text{II}}}} }}\left( {1 + \frac{{k_{{\text{I}}} }}{aRNh/K}} \right)} \right).$$
(9)

Thus, if \(\alpha\) is sufficiently larger than \(a\), no late spawning adult will oviposit in type-I habitats (Fig. 2). However, in either case of the partial or complete avoidance, if type-II habitats are of extremely low quality (\(u\) close to one), late spawning adults disfavor type-II habitats and select type-I habitats (Fig. 1a, b). The complete avoidance is more likely than the partial avoidance when the juvenile density (\(RN/K\)) is higher (Fig. 2a–d), the proportion of adults spawning in the early phase (\(h\)) is larger (Fig. 2e–h), and the proportion of type-I habitats (\({k}_{\text{I}}\)) is smaller (Fig. 2i–l), because these factors contribute to a higher density of juveniles hatching early in type-I habitats.

On the contrary, if \(\alpha\) is lower than the critical value, \({\alpha }_{\text{crit}}\), then late spawning adults prefer type-I habitats. In this case, the decrease of type-II habitat quality (larger \(u\)) increases the preference of late spawning adults toward type-I habitats (Fig. 1c, d); \({p}_{\text{L}}^{*}\) monotonically increases as \(u\) increases and reaches one when \(u\) is sufficiently large. Responses of \({p}_{\text{L}}^{*}\) to \(u\) may be weaker (Fig. 1c) or stronger (Fig. 1d) than those of \({p}_{\text{E}}^{*}\) (supplementary information). The stronger responses will occur if

$$\alpha < \alpha^{\prime\prime}_{{{\text{crit}}}} \equiv \frac{2a}{h}\left( {h - \frac{1}{2}} \right)$$
(8)

Noting that this critical value (\({\alpha }_{\text{crit}}^{{\prime}{\prime}}\)) is negative if \(h<1/2\), this condition can hold only when \(h\ge 1/2\)(Fig. 2e–h). In other words, the stronger responses can occur only when a larger proportion of adults oviposit in early than late phase. As \(h\) becomes larger, the stronger responses are more likely (Fig. 2g, h).

The change of baseline fitness in the late phase (\(\beta\)) does not affect the habitat selection of either early or late spawning adults. This is because \(\beta\) does not cause a fitness difference between type-I and II habitats in early and late spawning adults.

Discussion

Previous field observations and experiments have shown that spawning habitat selection by anuran adults involves complex decisions dependent on the densities of potential competitors and predators and changing abiotic conditions (Buxton and Sperry 2017). In contrast to accumulating empirical examples, there are few formal theories to examine how various biotic and abiotic conditions affect the spawning habitat selection of anurans. In this paper, we have developed a mathematical model to analyze how spawning timing, conspecific juvenile densities, and the quality of juvenile habitats can interactively affect the optimal spawning habitat selection of anuran adults.

An important prediction from our theory is that while early spawning adults consistently prefer superior habitats under relatively weak competition (Fig. 1a–d), late spawning adults can prefer inferior habitats (Fig. 1a, b). These patterns occur when the negative effect of intraspecific competition and cannibalism from early hatching juveniles (\(\alpha\)) is sufficiently strong (Fig. 2). In consistent with this prediction, spawning adults of Hyla chrysoscelis preferred to oviposit in experimental pools stocked with predatory fish rather than in fish-free pools when the fish-free pools were filled with conspecific eggs (Kraus and Vonesh 2010). Separate experiments also made similar observations for H. squirella and H. chrysoscelis (Binckley and Resetarits 2008). Moreover, other experiments show that late spawning adults evaluate the risk of conspecific antagonistic interactions on their juveniles and select spawning habitats accordingly, although these experiments do not directly test if adults can compare the risk of conspecific antagonistic interactions relative to habitat quality. Spawning adults of Rana japonica avoided experimental pools containing conspecific egg mass of advanced developmental stages but oviposited in pools with conspecific egg mass of unadvanced stages, suggesting that late spawning females selected habitats with a lower risk of cannibalism on their juveniles (Iwai et al. 2007). Similarly, adults of Pleurodema borellii chose to oviposit in artificial pools containing small conspecific tadpoles rather than those containing tadpoles of medium to large sizes (Halloy 2006).

Our theory also reveals circumferential conditions that make late-spawning adults choose inferior habitats. First, lower juvenile densities facilitate the selection of inferior habitats by late spawning adults (Fig. 2a–d), because lower juvenile densities translate to lower densities of late hatching juveniles (Eq. 7). Second, increasing the proportion of early spawning adults (\(h\)) is predicted to make the preference of late spawning adults extreme, resulting in a stronger preference for either type-I or type-II habitat (the regions for moderate preference shrink in Fig. 2e–h). These conditions might help explain the variation of oviposition habitat preference among late spawning adults observed in empirical studies (Buxton and Sperry 2017).

According to our theory, even when the negative effects of early-hatching juveniles on late-hatching juveniles are so large that late-spawning adults would avoid superior habitats containing elder juveniles, the adults may nonetheless oviposit in those superior habitats if the quality of inferior habitats is too low (large \(u\) in Fig. 1c, d). Although no experiment seems to have considered such an ultimate situation, the prediction could be tested in an experiment that provides spawning adults with a choice among habitats containing conspecific offspring and habitats of significantly low quality. Considering that spawning adults can perceive the cost of avoiding competition/predation in superior habitats (Resetarits and Wilbur 1989; Buxton et al. 2017), those adults might choose the superior habitats if the alternative inferior habitats are of substantially poor quality.

While our theory considers multiple factors known to affect spawning habitat selection by anurans (i.e., habitat quality, spawning timing, and competitor and predator densities), additional factors might affect the theoretical predictions. For example, our theory does not consider the positive effects of high conspecific densities. Sharing a juvenile habitat with a large number of conspecific eggs/juveniles can be advantageous, if communal egg masses benefit from thermal insulation or if large abundances satiate predators (Doody et al. 2009). Such positive effects may prompt spawning adults to choose habitats containing more conspecifics. Lowering \(a\) (or even reversing the sign) in our model might imitate such effects. Additionally, in our model, the relative difference in habitat quality between early and late spawning phases does not affect spawning habitat selection. This is because the quality difference between phases is assumed to affect type-I and II habitats equally. If the quality difference between type-I and II habitats changes between spawning phases, then it could affect spawning habitat selection. Moreover, habitat quality can affect strength of intraspecific competition and cannibalism; for example, the antagonistic interactions might be severer in resource-limited inferior habitats (Jefferson et al. 2014; Gould et al. 2021). Also, larger females with high fecundity can be more likely to spawn early (Tejedo 1992). Incorporating these scenarios into the model could modify the model predictions.

Finally, our model might be more generally applicable to other organisms than anurans, such as some insects, in which adults provide no parental care but select oviposition habitats to increase the survival and growth of their juveniles (Resetarits 1996; Refsnider and Janzen 2010). In the butterfly Anthocharis scolymus, for example, seasonally-delayed oviposition increases the risk of egg cannibalism by early hatching larvae (Kinoshita 1998). In the damselfly Lestes sponsa, which has an extended spawning season (Helebrandová et al. 2018), larvae are involved in size-mediated intra-cohort competition and cannibalism (Sniegula et al. 2019). Our theory suggests that late ovipositing adults in such non-anuran organisms may also choose natal habitats having fewer conspecific offsprings at the cost of habitat quality.