^{1}

^{2}

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Conceived and designed the experiments: TN. Performed the experiments: TN. Analyzed the data: TN. Contributed reagents/materials/analysis tools: TN. Wrote the paper: TN.

The author has declared that no competing interests exist.

It has been suggested that when juveniles and adults use different resources or habitats, alternative stable states (ASS) may exist in systems coupled by an ontogenetic niche shift. However, mainly the simplest system, i.e., the one-consumer–two-resource system, has been studied previously, and little is known about the development of ASS existing in more complex systems. Here, I theoretically investigated the development of ASS caused by an ontogenetic niche shift in the presence of multiple resource use. I considered three independent scenarios; (i) additional resources, (ii) multiple habitats, and (iii) interstage resource sharing. The model analyses illustrate that relative balance between the total resource availability in the juvenile and adult habitats is crucial for the development of ASS. This balance is determined by factors such as local habitat productivity, subsidy inputs, colonization area, and foraging mobility. Furthermore, it is also shown that interstage resource sharing generally suppresses ASS. These results suggest that the anthropogenic impacts of habitat modifications (e.g., fragmentation and destruction) or interaction modifications (e.g., changes in ontogeny and foraging behavior) propagate through space and may cause or prevent regime shifts in the regional community structure.

Many animals change their resource or habitat use during the course of individual growth; such a change is known as ontogenetic niche shift

The ecological consequences of ontogenetic food-web coupling have been investigated in only a few recent theoretical studies

Previous theoretical studies have only considered one consumer-two resource systems

With the term “multiple resource use,” in this study I define that juveniles and/or adults use more than two resources at each stage. The resource use may be of different types. In the present study, I consider the following three independent scenarios for model development. In the first scenario, I assume that juveniles and adults have “additional resources” (

Throughout the modeling, I follow the previous studies

First, I explore the model in which the juveniles and adults have additional resources (_{h}_{,i} (_{h}_{h}_{h}_{,i} and _{h}_{,i} are the intrinsic growth rate and carrying capacity, respectively of the _{h}_{h}_{,i} and _{h}_{,i} are the consumption rate and energy conversion efficiency of the _{h}

I examine the multiplicity of coexistence equilibria by performing zero-net-growth isocline (ZNGI) analysis. Consider the equilibrium state in which the juvenile and adult animals coexist with all resources. Using equations 1a and 1b, I obtain _{J}_{,i}^{*} = _{J}_{,i}(1−_{J}_{,i}_{J}^{*}/_{J}_{,i}) and _{A}_{,i}^{*} = _{A}_{,i}(1−_{A}_{,i}_{A}^{*}/_{A}_{,i}), respectively (asterisks denote equilibrium quantities). Substituting these expressions into _{A}_{J}_{A}_{A} and ZNGI_{J}, respectively. The intersections of the two ZNGIs determine the coexistence equilibria. One solution is always trivial; for this solution, _{J}^{*} = _{A}^{*} = 0. The point in the analysis is that _{J}^{*} and _{A}^{*} are expressed as upward-convex quadratic functions of each other. Therefore, at most three coexistence equilibria are observed when ASS exist: one is an unstable equilibrium and the other two are stable ones (stable equilibrium point or stable periodic orbits;

The coexistence equilibria are obtained by solving a cubic equation _{J}^{*}) = _{1}(_{J}^{*})^{3}+_{2}(_{J}^{*})^{2}+_{3}_{J}^{*}+_{4} = 0, which is derived by substituting ZNGI_{A} in ZNGI_{J} (note that one solution is trivial). A necessary condition for the existence of ASS is that this equation has three positive solutions. Using the discriminant of a cubic equation, this condition is given as_{1}>0. Parameter space for ASS can be numerically evaluated by using inequalities 3a and 3b.

Here, I briefly show the parameter-dependence of the occurrence of ASS. For presentation, I simply assume that the juveniles and adults have two resources (i.e., _{h}_{J}_{,2} or _{A}_{,2} while keeping the other parameters fixed. The ZNGI analysis shows that ZNGI_{A} (or ZNGI_{J}) shifts to the upper right with an increase in _{J}_{,2} (or _{A}_{,2}) in the space of _{J}^{*} and _{A}^{*} (left or center panel in _{A}^{*}/∂_{J}_{,2}>0 (or _{J}^{*}/_{A}_{,2}>0). These behaviors of the ZNGIs indicate that ASS exist when both _{J}_{,2} and _{A}_{,2} are sufficiently large but not when they differ considerably. This is illustrated by the analytical approach using inequalities 3a and 3b (right panel in

(A) additional resources, (B) multiple habitats, and (C) interstage resource sharing. In each scenario, the left and central columns show the results of ZNGI analysis. The solid and dotted lines represent ZNGI_{A} and ZNGI_{J}, respectively. The black lines are for the default parameter settings as described below. The blue and red lines represent ZNGI when one juvenile- or adult-specific parameter is increased. The solid and dotted arrows roughly denote the shift direction of ZNGI_{A} and ZNGI_{J}, respectively, with an increase in the corresponding parameter. The solid and open circles represent stable and unstable equilibria, respectively. The blue and red circles indicate the intersections on a ZNGI of the same color. The right panel shows the analytical results for ASS in a corresponding two-parameter space. ASS exist in the green region. (A) _{h}_{J}_{,2} = 10, 20, or 30 and _{A}_{,2} = 10; center: _{J}_{,2} = 10 and _{A}_{,2} = 10, 20, or 30; right: _{J}_{,2} and _{A}_{,2} are variables. (B) _{h}_{J}_{A}_{J}_{A}_{J}_{A}_{h}_{J}_{,A} = 0.01, 0.05, or 0.1 and _{A}_{,J} = 0.01; center: _{J}_{,A} = 0.01 and _{A}_{,J} = 0.01, 0.05, or 0.1; right: _{J}_{,A} and _{A}_{,J} are variables. The other parameter values are set _{h}_{h}_{,i}) = 1, _{h}_{h}_{,i}) = 0.1, _{h}_{h}_{,i}) = 0.5, and _{h}_{h}_{,i}) = 0.1.

Next, I consider the situation where the juveniles and adults can colonize several habitats, where they exploit one resource (_{h}_{h}_{,i} = _{h}_{h}_{,i} = _{h}_{h}_{,i} = _{h}_{h}_{,i} = _{h}_{h}_{,i} = _{h}_{h}_{,i}^{*} = _{h}^{*} and _{h}_{,i}^{*} = _{h}^{*}.

From equations 4a and 4b, I obtain _{J}_{,i}^{*} = _{J}_{,i}(1−_{J}_{,i}_{J}^{*}/_{J}_{,i}) and _{A}_{,i}^{*} = _{A}_{,i}(1−_{A}_{,i}_{A}^{*}/_{A}_{,i}), respectively. Substituting these expressions into _{A}_{J}_{A}_{J}_{A}_{J}_{A}_{A} and ZNGI_{J}: ZNGI_{A} and ZNGI_{J} shift upward and to the left, respectively, with an increase in _{J}_{J}^{*} and _{A}^{*} (left panel in _{A}_{J}_{A}_{h}

This model should be extended to include the spatial heterogeneities in stage-specific local environmental conditions. Here, I only briefly present the preliminary numerical results. For simplicity, I assume that both the juveniles and adults have two habitats, and introduce environmental heterogeneity as a difference in productivity between the two juvenile habitats. The results indicate that the system has at least three ASS for some parameter settings (

Finally, I assume that the juveniles and adults can utilize the major resources of the other life-history stage (_{h}_{,i} (_{i}_{h}_{,i} is the conversion efficiency for _{h}_{,i}.

From equations 6a and 6b, I obtain _{J}_{,i}^{*} = _{J}_{,i}(1−_{J}_{,i}_{J}^{*}/_{J}_{,i}) and _{A}_{,i}^{*} = _{A}_{,i}(1−_{A}_{,i}_{A}^{*}/_{A}_{,i}), respectively. Substituting these expressions into _{A}_{J}_{A}_{A} and ZNGI_{J} are fractional functions of each other; their numerator and denominator are upward-convex quadratic and linearly increasing functions, respectively. Here, I focus on the effect of varying _{J}_{,A} or _{A}_{,J} on the development of ASS. In the space of _{J}^{*} and _{A}^{*}, ZNGI_{A} and ZNGI_{J} generally shifts to the lower left and downward with an increase in _{J}_{,A} or _{A}_{,J}, respectively (left and center panels in _{A} and ZNGI_{J} can be represented by a cubic equation (not shown). The analytical results illustrate that ASS are generally suppressed when either _{J}_{,A} or _{A}_{,J} or both are large (right panel in _{J}_{,A} and _{A}_{,J} are very large (right panel in

In this study, I theoretically investigated the development of ASS resulting from ontogenetic habitat coupling in the presence of multiple resource use in different scenarios (

Previous theoretical studies, in which one-consumer-two-resource systems were considered, suggest that the relative balance of the juvenile and adult habitat productivities is crucial for the development of ASS

I also showed that interstage resource sharing generally suppresses ASS (

My models were developed for the specific purpose of analytically identifying the conditions for ASS in the presence of multiple resource use. I therefore purposely formulated simple analytical models without incorporating additional factors affecting population dynamics. To better understand the occurrence of ASS in reality, therefore, the models need to be modified or extended as in the following examples. First, one may be concerned with the case in which trophic interactions are nonlinear. If the functional forms are nonlinear due to a long handling time or strong interference competition, it decreases the likelihood of ASS (see

Finally, I like to emphasize that ontogenetic habitat coupling may produce more diverse and complex community dynamics in the presence of multiple resource use. It is known that a consumer of multiple resources may drive some of the resources extinct, or to very low levels, via apparent competition

In conclusion, the present study demonstrated that the development of ASS is critically affected by the food-web structure at lower trophic levels; the development is determined by various factors such as spatial resource distribution, colonization area, consumer mobility, and stage-specificity of resource utilization. Currently, there are increasing concerns about anthropogenic impacts of eutrophication

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I thank Hideyuki Doi and two reviewers for helpful comments on this manuscript.