^{1}

^{*}

^{2}

Seed dispersal of juniper and pinyon is a process in which frugivorous birds play an important role. Birds either consume and digest seeds or carry and cache them at some distance from the source tree. These transported and settled seeds can be described by a dispersal kernel, which captures the probability that the seed will move a certain distance by the end of the process. To model active seed dispersal of this nature, we introduce handling time probabilities into the dispersal model to generate a seed digestion kernel. In the limit of no variability in handling time the seed digestion kernel is Gaussian, whereas for uniform variability in handling time the kernel approaches a Laplace distribution. This allows us to standardize spatial movement (diffusion) and handling time (peak settling rate) parameters for all three distributions and compare. Analysis of the tails indicates that the seed digestion kernel decays at a rate intermediate between Gaussian and Laplace seed kernels. Using this seed digestion kernel, we create an invasion model to estimate the speed at which juniper and pinyon forest boundaries move. We find that the speed of seed invasion corresponding to the digestion kernel was faster than seeds resulting from Laplace and Gaussian kernels for more rapidly digested seeds. For longer handling times the speeds are bounded between the Laplace (faster) and Gaussian (slower) speeds. Using parameter values from the literature we evaluate the migration potential of pinyon and juniper, finding that pinyon may be able to migrate up to two orders of magnitude more rapidly, consistent with observations of pine migration during the Holocene.

Forest boundaries change over time, and in favorable climates can expand as tree seeds spread beyond the range of the forest and germinate into new trees. Seeds may spread in a variety of ways. Common seed dispersal agents include wind, transportation in water, and transportation via birds and animals (either through being consumed and digested or being carried and cached). Because the diet of birds and some animals is often made up of fleshy-fruited plants, the pattern of seed dispersal and activities of vertebrate dispersers are closely related [

A case in point is seed dispersal and forest migration in two southwestern tree species: pinyon (Pinus monophylla) and juniper (Juniperus osteosperma). In both species birds play a major, but different, role. Junipers produce seeds which are available most of the winter, and consequently many birds (particularly American robins, Turdus migratorius, and cedar waxwings, Bombycilla cedrorum) consume juniper berries [

By contrast, pinyon seed dispersal by Clark’s nutcracker (Nucifraga columbiana) and pinyon jays (Gymnorhinus cyanocephalus) occurs primarily through seed caching in summer and fall, when the cones mature. Some seeds are consumed immediately, but the majority is placed in a sublingual pouch and carried several kilometers to remote cache sites, where they are buried in small groups [

Variation in climate also influences the expansion of pinyon-juniper (P-J) woodland via impacts on germination and survival rates. The abundance of summer rainfall and the warming in the winter and spring have caused P-J boundaries to shift northwards [

To model the spread of the seeds, we assume that seed cachers or frugivorous animals collect seeds (through consumption in the case of juniper berries or collection of pinyon seeds to cache at a distance) and then follow a random walk, using the modeling framework introduced by [

In this paper we numerically calculate the SDK based on PDFs of seed handling times. Once the kernel is determined, we will use it in a generic IDE population model to estimate invasion speeds and compare with speeds generated from Gaussian and Laplace kernels, which are limiting cases. Surprisingly, in some parameter regimes the SDK yields more rapid invasion speeds than either Laplace or Gaussian kernels. Predictions for juniper and pinyon migration rates will be generated using literature values for parameters. We find that pinyon has much higher potential to find and occupy new niches than juniper, consistent with observations of Holocene range expansion for the two species.

We begin with a common model of dispersal and settling of propagules, (any material that is used for propagating an organism) introduced by [

In this model

An important modeling point is that, to be consistent with mechanisms of seed handling by vertebrates, the hazard function,

In this distribution the constant b scales the mean digestion time of seeds while a is a normalization constant (not free, since it depends directly on the other three parameters so that

We integrate the PDE directly and then approximate time integrals using the trapezoid rule. To begin, let

Equation (1) becomes

An integrating factor of

Using Equation (6) in the model (2) then we get

Numerical approximations are then calculated using the trapezoid rule for numerical integration. Solutions generated this way were cross-checked against the (much) more time-consuming finite difference solution of (1) and (2) (see Appendix A) to ensure accuracy. For the same size of time steps we found that direct quadrature of integrals in (7) was substantially more accurate (and rapid) than solution of the PDEs using finite differences.

We wish to compare the SDK with the Gaussian and Laplace seed. The question is how to standardize the three kernels for comparison? Below we will show that the Laplace and Gaussian kernels arrive from different limiting choices for

For convenience, we assume

We will call this time

Then

This gives

To generate a Laplace kernel we assume that the distribution of seed settling is a step function defined by

As is shown in Appendix B, the solution to (1) and (2) with the step function defined in Equation (11) is approximately the Laplace kernel

After this standardization, the Gaussian kernel (10) and the Laplace kernel (12) are ready for comparison with the seed digestion kernel.

Here, we approximate the tail of the SDK using the steepest descent method [

Define the exponent in (7) as

The critical point of the function

with respect to t and evaluating at

Let us suppose

Equations (15) and (16) can be used with the generalized steepest descent theorem to approximate (7), giving

Analyzing K as

To determine how the shape of the SDK affects rates of invasion, the kernels must be imbedded in a population model. Below we present a simplification of the model used by [

where N_{t} represents the population density of adults in generation t. The function

Here M is the maximum number of surviving seeds, g is the germination rate, and

and the integral represents the total number seeds arriving at location x from all possible locations, y. Therefore, the first term on the right hand of the invasion model (18) predicts the distribution of new trees depending on the available sources and the second term provides the surviving number of old trees so that the total is the population of trees in the next generation.

To analyze invasion speeds for models like (18), we follow the analysis of [

with

Combining Equations (19) and (20)

Plugging this into Equation (18),

Taking only leading order terms,

where

Writing the convolution of Equation (23) in terms of an integral, we have

where the moment generating function, M(u), is defined by

Differentiating Equation (24) with respect to u and setting to zero to find the extremal invasion speed gives

Using Equation (26) to eliminate c in (24) gives

To find the invasion speed,

The mean digestion time scaling parameter b plays a major role in determining seed dispersal. Changing the value of b generates different shapes of solutions (see

We would like to characterize the shape of the tail of the SDK and place it in the context of the well-known Gaussian (10) and Laplace kernels (12). The exponents of the exponential functions for both kernels determine

the shapes of the corresponding tails. To analyze the tails of these kernels, we consider large x and assume other parameter values are bounded. The dominant terms in the exponents of the Gaussian and Laplace kernels are

than the tail of the Laplace kernel when

In the case of the most slowly-decaying PDF of seed handling times,

where

and therefore

The exponents of Gaussian kernel and Laplace kernels are

and

Observing

we conclude that the tail of Gaussian kernel decays to zero most rapidly while the tail of Laplace kernel is the slowest. The tail of the SDK is intermediate between the other two.

We have not chosen scales for generation time, population density and space yet. To compare the speeds of invasion from the population IDE (18), we may therefore, without loss of generality, choose mortality

There is a strong relationship between the characteristic handling time, b, and invasion speed, c. The longer it takes to digest a seed, the faster forest migration. For small b, the SDK invasion speed is higher than the speeds corresponding to the Gaussian and Laplace kernels. On the other hand, for bigger b values, the speed of invasion with the Laplace kernel is fastest, the speed with a Gaussian kernel is slowest and the speed corresponding to the SDK stays between the other two, as might be expected from comparing tails.

As

To quantify invasion speeds in terms of yearly distance covered for both pinyon and juniper, we need specific data such as the mean generation time (G), mean dispersal space step

where G is the generation time (duration from seedling to getting matured tree for producing seeds), n is the initial number of trees and N is the total number of trees at the end of the study. To estimate the diffusion rate D for birds, we use

where

In order to calculate yearly invasion speed, c, we need to rescale both space and time, since we fixed D = 1 (equivalent to nondimensionalized space) in the numerical calculations. Additionally, each step in our IDE is a generation, which must be scaled back to years for comparison purposes. Assuming the dimensional diffusion rate, D, is in m^{2} per minute and the mean handling time

Parameter | Juniper | Reference | Pinyon | Reference |
---|---|---|---|---|

Generation time (G) | 50 yrs | Li et al. (2011) | 20 yrs | Suzan-Azpiri et al. (2002) |

Mean dispersal space step (λ) | 55 m | Chavez-Ramirez and Slack (1994) | 4500 m | Vander Wall and Balda (1977) |

Mean dispersal time step (τ) | 4 min | Chavez-Ramirez and Slack (1994) | 22.5 min | Vander Wall and Balda (1977) |

Diffusion (D) | 189.1 m^{2}/min | Calculated | 225,000 m^{2}/min | Calculated |

Mean handling time (b) | 14.9 min | Holthuijzen and Adkisson (1984) | 52.5 min | Vander Wall and Balda (1977) |

Reproductive rate (R_{0}) | 1.17 /gen | Tausch and West (1988) | 2.04/gen | Tausch and West (1988) |

Mortality (ω) | 0.0004 | Shaw et al. (2005) | 0.00155 | Shaw et al. (2005) |

Now we turn to specific parameter values for pinyon and juniper.

For pinyon we use a generation time G ~ 20 years from [_{0} we refer to [_{0} = 2.04/generation for pinyon from equation (33). [

λ = 4500 meters and use τ = 22.5 minutes, giving

three phases. Nutcrackers spend 45 minutes collecting seeds to fill their pouch, 15 - 30 minutes to travel to the caching area and 5 - 10 minutes to cache all seeds carried in their pouch. Averaging and summing, we estimate

seed mean handling time to be

timate annual mortality at 0.08% - 0.23% for common pinyon. Taking the mean and converting to a rate per generation gives ω = 0.00155. Taken together, these parameters give a minimum speed of 518.97 m/year and a maximum of 946.1 m/year for pinyon, with an average of 773.31 m/year.

On the other hand, for juniper we use a generation time G ~ 50 years from [_{0} we follow [_{0} = 1.17/generation for juniper from Equation (33). [^{2}/min. [

therefore use a handling time

estimate annual mortality at 0.01% - 0.07% for Utah juniper (Juniperus osteosperma). Taking the mean and converting to a rate per generation gives ω = 0.0004. Using these parameter ranges we find that juniper spreads with minimum speed 0.42 m/year and maximum speed 7.3 m/year with an average of 3.3 m/year, two orders of magnitude more slowly than pinyon.

These results match up well with what is known about these two species and their relative movements during the Holocene. Juniper seems to have been present in the Great Basin area for at least 30,000 years, based on evidence from fossilized packrat middens [

Mechanisms of plant migration vary based on the source plant and the dispersal process delivering seeds to new locations for germination. Juniper berries mainly disperse after being eaten by vertebrates who deposit seeds after digestion. Birds, particularly robins, may be the biggest dispersers. Seeds of pinyon trees, on the other hand, are commonly spread while animals cache, and corvids (jays and nutrcrackers), which cache at large distances, are the largest contributors. In this paper, we introduced a PDF of seed-handling to reflect the effects of digestion/caching on dispersal of pinyon and juniper seeds. We connected this distribution to hazard functions or failure rates in an existing random-walk dispersal model to determine a seed digestion kernel modeling the probable location of seeds after active dispersal. As expected, if birds or animals take more time to handle seeds, those seeds are dispersed further away from the source tree. While no closed-form solution for the SDK is available, it is easy to calculate numerically (and would only have to be calculated once, in advance, for implementation in an IDE model for population invasion).

To evaluate migration potential for pinyon and juniper we introduced an IDE model with competition among seedlings, which is appropriate for desert-adapted trees in the xeric environment of the American Southwest. The SDK was compared with well-known Laplace and Gaussian kernels (L(x) and G(x)). After standardizing the associated PDFs for handling time, the speed of invasion for the SDK was the fastest for shorter handling times (rapidly digesting seeds). As handling times increased, however, the speeds for the SDK fell between the Laplace kernel’s (faster; based on an assumption of constant seed deposition) and the Gaussian kernel’s (slower; based on the assumption of instantaneous seed deposition), as would be expected from the relative behavior of the tails.

Using the SDK and median parameter values estimated from the literature it turned out that pinyon has migration potential at least two orders of magnitude larger than juniper due to avian dispersal. Along with changing temperatures and diminishing moisture levels the favorable environment for P-J is moving northwards through Utah. Over time, these trees will not be able to survive in the southern limits of their current habitat. The large migration potential of pinyon means that it is most likely to occupy new habitats opening to the north.

Of course, juniper already occupies much of the available northern habitat, and with longer generation times and much stronger adaptation to variable moisture regimes juniper can be expected to flourish in northern Utah for the foreseeable future. Moreover, juniper may have much higher migration potential than our analysis indicates. For the slower juniper we can probably not ignore mammalian dispersers [

The largest factor ignored in our study is spatial variability. As [

The authors would like to thank Tom Edwards and Jacob Gibson for discussion of life-history traits of pinyon and juniper and assistance with background literature. Luis Gordillo, Martha Garlick and USU’s MathBio group gave RCN much helpful feedback. RCN was partially supported by a grant from USU’s School of Graduate Studies. JAP would also like to thank the NSF for support under DEB grant 0918756 as well as the Western Wildland Environmental Threat Assessment Center (WWETAC).

Ram C.Neupane,James A.Powell, (2015) Mathematical Model of Seed Dispersal by Frugivorous Birds and Migration Potential of Pinyon and Juniper in Utah. Applied Mathematics,06,1506-1523. doi: 10.4236/am.2015.69135

We will use two techniques to solve the dispersal model given by Equations (1) and (2) with corresponding seed digestion rate given by the Equation (3). First, we will solve this PDE numerically using finite difference approximations

where

Using these discretizations and approximating

with variance σ chosen to be very small, the system (1) and (2) can be solved numerically provided

The Laplace kernel used for comparison in the main text arises from solving Equations (1) and (2) with a con-

stant hazard function

lace kernel with the seed digestion kernel with step function

approximation of the systems (1) and (2) with a step function failure rate

To calculate the difference between the Laplace and actual kernels, we first find the error in P and use it to calculate the error in S from the models (1) and (2). We denote the actual solution for P (with stepped hazard function) as

equal when

The solution for

when

when

The error in P as

Thus, the error in S is

Finally we must calculate the error in the kernel (

We use the trapezoid rule to approximate the integral and note that we have not chosen scalings of time and space, so without loss of generality D = 1 and b = 20. In a spatial domain −30 < x < 30 the estimated