Rapid changes in seed dispersal traits may modify plant responses to global change

Abstract When climatic or environmental conditions change, plant populations must either adapt to these new conditions, or track their niche via seed dispersal. Adaptation of plants to different abiotic environments has mostly been discussed with respect to physiological and demographic parameters that allow local persistence. However, rapid modifications in response to changing environmental conditions can also affect seed dispersal, both via plant traits and via their dispersal agents. Studying such changes empirically is challenging, due to the high variability in dispersal success, resulting from environmental heterogeneity, and substantial phenotypic variability of dispersal-related traits of seeds and their dispersers. The exact mechanisms that drive rapid changes are often not well understood, but the ecological implications of these processes are essential determinants of dispersal success, and deserve more attention from ecologists, especially in the context of adaptation to global change. We outline the evidence for rapid changes in seed dispersal traits by discussing variability due to plasticity or genetics broadly, and describe the specific traits and biological systems in which variability in dispersal is being studied, before discussing some of the potential underlying mechanisms. We then address future research needs and propose a simulation model that incorporates phenotypic plasticity in seed dispersal. We close with a call to action and encourage ecologists and biologist to embrace the challenge of better understanding rapid changes in seed dispersal and their consequences for the reaction of plant populations to global change.

delineates the parameters and variables in the model. STEP ONE (Seed production): Seeds of type i per area (density of seed production, in effect) produced are given by: Remark 1. In (1), i could be discrete or continuous. In the simplest (discrete) case i = 1, 2, ..., but there could be several types or even a continuous distribution of types. Table S1: Biological interpretations of parameters and variables.
P(x,t) adult plant density at location x and time t. b(x,t) level of energy (resources) available for making seeds (per captia) at (x,t). γ i cost of producing seeds of type i, i = 1, 2. α i (x,t) fraction of seeds that are type i at (x,t), i = 1, 2. S i (x,t) density of seeds arriving at x, of type i, at time t. k i (x, y) dispersal kernel for seeds of type i. Ω spatial region. Table S1, b(x,t) could depend on P(x,t). For α i (x,t), the simple case of two seed types has α 1 = α and α 2 = 1 − α, 0 ≤ α ≤ 1. Again more complicated situations are possible,

Remark 2. In
i.e., α i (x,t) might depend on b(x,t) and/or P(x,t). Such dependence would be where we expect plasticity to come into play. In general, for multiple type seeds (i ≥ 2), α i (x,t) would be analogous to a probability distribution in that Dispersal is modeled by a spatial integral of the product of the produced seeds of type i from Step one and the dispersal kernel k i (x, y), the probability of an individual moving from location y to x. In other words, the seed density of type i in the next generation, S i (x,t + 1) arises by tallying arrivals at x from all possible locations y.

STEP THREE (Establishment):
This could be done in various ways. Here we have thought about it in terms of a "metapopulation" type of competition for space, but in continuous space. (The key idea here is that population growth comes from seeds colonizing empty space, and thus is tightly linked to dispersal. In this way, the model is more akin to metapopulation models than reaction-diffusion models where dispersal and growth are independent processes.) Parameters and variables included in Step Three appear in Table S2. K(x) density of suitable sites at location x. σ (x,t) fraction of adults surviving from time t to t + 1, at location x. g i (x,t) probability of germination and growth of seed type i. S(x,t) density of germinated seeds at location x, at time t. F(S) fraction of available space that will be colonized by a rain of seeds at level S. If σ (x,t) is defined as in Table S2, then at time t + 1 the fraction of the total number of sites at location x that is occupied by adults would be , so the fraction of sites available . These would be colonized by the incoming seeds.
and F(S) be as given in Table S2. We would want with a general assumption that F increases in S. We take a Monod/Holling II form for analytic simplicity: A Holling I form might also be reasonable in some situations. Note that we can describe S as wherek is a complex kernel built from the k i ' s and other parameters. Then At the end of Step Three, one has The case of annuals (σ (x,t) = 0) is probably simpler than other cases in some ways.
To elaborate the model to compare competing plasticity strategies, we employ the superscript j to denote sub-populations of the species using different plasticity strategies. For each sub-population we calculate up to equation (6) as before, with all parameters and variables indexed by j, arriving at S j for sub-population j( j = 1, 2, ..., J). (We might want b(x,t) to depend on (P 1 (x,t), .., P J (x,t))). Then, instead of (7), we would get equations for j = 1, 2, ..., J. Here we take or something similar, (with S j = S j (x,t + 1).) (Recall from (6) that S j includes kernels coefficients g j i that could describe competitive strength, and coefficients α j i that describe plasticity.) Remark 4. In (6), we could think of α = α(x,t, i) as corresponding to a continuous probablility distribution in i for each (x,t), and replace ∑ i with I 2 I 1 ( )di.
Conclusion: Employing a model such as (8), (9) with two populations (J = 1, 2) allows us to use pairwise invasibility analysis to study the evolutionary stability of plasticity patterns in dispersal, or related questions.

Numerical results
To illustrate the effects of the plasticity α on the rates of spread and persistence in a spatial heterogeneous environment, we simplify the spatial heterogeneity by considering two types of periodically fragmented patches (Shigesada et al., 1986), denoted as 'good' and 'bad' according to m(x) := b(x)K(x). Next, we denote by l 1 the length of a bad patch and by l the period of the landscape, so that l − l 1 is the length of a good patch. Let F(s) = s s + 1 , p(x,t) = P(x,t) K(x) and G i := g i γ i , which measures the overall survival ability for type i (i = 1, 2), α 1 = α and α 2 = 1 − α. Then the model (7) becomes wherek(x, y) Numerically, we first choose the Gaussian kernels k i (x) = 1 as an example to explore the following scenario: there are only two types of seeds: type 1 and type 2, where type 1 refers to big seeds and type 2 represents small seeds. More specifically, type 1 (big) seeds are assumed to be fewer, but with strong germination properties, and governed by a short range kernel, while type 2 (small) seeds are assumed to be more abundant, but with poorer germination, and subject to a wide ranging kernel. Therefore, this implies γ 1 > γ 2 , g 1 > g 2 , δ 1 < δ 2 .
How does spread rate depend on α ? For our model, because of the form F(s), the overall dynamical behaviors of our model in a periodic habitat would be analogous to that of the classic Beverton-Holt model, that is, the positive solutions of the model system converge to either zero or a unique positive periodic steady state locally uniformly, in other words, colonizing the local region in a periodic (oscillated) fashion when it could survive. (Fig. S1(a) and S1(b)). Recall that the spread rate is an asymptotic rate (slope) defined as the limit of front location x(t) over t as t goes to infinity (Kawasaki and Shigesada, 2007). Since G 1 = G 2 , small seeds have a larger spread rate due to the associated wider ranging kernel (see Fig. S1(c)). Fig. S2 shows the spread rates under different G i (i = 1, 2). If G 1 ≤ G 2 , the largest spread rates would be the single type with small seeds (see Fig. S2(a)). But if G 1 is greater than G 2 appropriately, the plasticity α > 0 might give a larger spread rate with the mixed types of seeds(see Figs. S2(b) and S2(c)). If G 1 is sufficiently larger than G 2 , the largest rates would be the single type with big seeds (see Fig. S2(d)). In any case, we could have c(α) ≥ αc(0) + (1 − α)c(1). A similar result for system (10) with Laplace Fig. S3 and S4.   Figure S2: The spread rate c is a function of α with different G i (i = 1, 2) and the same parameter values as in Fig. S1.     Figure S4: The spread rate c is a function of α with different G i and the same parameter values as in Figs. S1 and S3 with the same variance.