James W. Haefner

Edge effects in computer models of spatial competition

Abstract Spatially explicit models of biological growth processes make assumptions about the interactions near the edges of the finite plots modeled. The most realistic approach is to embed the plot of interest in a larger space, but this is computationally costly and only moves the edge further away. To determine the bias introduced by using computationally efficient but less realistic methods, we used Monte Carlo simulation to compare four approaches to modeling the effects of boundaries on sessile organisms growing into asymmetric shapes under crowded conditions in finite regions. The four algorithms compared were: (a) embedding the modeled area in a space larger than the plot, (b) reflecting the area over the boundaries, (c) mapping the area onto a torus (translation), and (d) linearly expanding (extrapolating) the proportion of occupied area outside from that occupied inside. We studied the behavior of each algorithm in different ecological situations that varied the intensity of interactions and maximum sizes of individuals. Asymmetric growth due to interactions between individuals was modeled, and six different indices of individual size and shape were compared. When spatial interactions were intense, the embedded method was significantly different from other algorithms in almost all comparisons. Over all treatments and dependent variables, the torus was different from the embedded method in 18 of 30 comparisons. The reflection and linear expansion algorithms were approximately equal in their disagreement with the embedding space method: 28 and 27 comparisons, respectively, were different out of 30 comparisons. Dependent variables were ranked by the numerical differences between the embedding and non-embedding methods: measures of individual growth asymmetry produced the greate differences; measures of interactions between individuals produced intermediate differences; measures of individual size and percent cover were the least different. The relative numerical differences between the means of the variables produced by the non-embedding algorithms and the embedding method ranged from 60% to 0%, but 1 2 of these comparisons were within 5% of the embedding method estimates.

Seed Dispersal in Heterogeneous Environments: Bridging the Gap between Mechanistic Dispersal and Forest Dynamics Models

Seed dispersal is an important determinant of vegetation composition. We present a mechanistic model of seed dispersal by wind that incorporates heterogeneous vegetation structure. Vegetation affects wind speeds, a primary determinant of dispersal distance. Existing models combine wind speed and fall velocity of seeds. We expand on them by allowing vegetation, and thus wind profiles, to vary along seed trajectories, making the model applicable to any wind‐dispersed plant in any community. Using seed trap data on seeds dispersing from forests into adjacent sites of two distinct vegetation structures, we show that our model was unbiased and accurate, even though dispersal patterns differed greatly between the two structures. Our spatially heterogeneous model performed better than models that assumed homogeneous vegetation for the same system. Its sensitivity to vegetation structure and ability to predict seed arrival when vegetation structure was incorporated demonstrates the model’s utility for providing realistic estimates of seed arrival in realistic landscapes. Thus, we begin to bridge mechanistic seed dispersal and forest dynamics models. We discuss the merits of our model for incorporation into forest simulators, applications where such incorporation has been or is likely to be especially fruitful, and future model refinements to increase understanding of seed dispersal by wind.

Inbreeding depression effects on extinction time in a predator—prey system

SummaryTraditional methods of assessing population viability ignore both genetic—demographic interactions as well as community level dynamics. We address these deficiencies by presenting a model that investigates the effects of predation on a prey population experiencing inbreeding depression. Beginning with a simple Lotka—Volterra predator—prey system, we rewrite prey per capita mortality as a function of inbreeding. Inbreeding varies as a function of population size. Using computer simulation, we find that prey extinction times are inversely related to the level of inbreeding depression with and without predation. For all but very low levels of inbreeding depression, predation appreciably reduces persistence time. At moderate levels of inbreeding, predators go extinct before prey. When migration is introduced at low and moderate rates, persistence times only improve for those populations with low inbreeding depression measures. At a higher migration rate, persistence times are lengthened for low and moderately depressed prey populations. Increasing birth rates produce a visible, though noisy, trend towards increased times to extinction for low to moderate levels of inbreeding.

Proximate cues for predator searching: a quantitative analysis of hunger and encounter rate in the ladybird beetle, Coccinella septempunctata

Many mobile predators use different foraging behaviours depending on whether prey are clumped or uniformly distributed in space. It is believed that predators can increase foraging efficiency on patchily distributed prey if they alternate intensive searching (slow speeds and frequent, acute turns) with extensive searching (high speeds and infrequent, shallow turns). We used Fourier transforms, fractal dimension analysis and traditional single-value measures of movement to determine which proximate cue caused these changes in behaviour: frequency of encounter or degree of satiation. Over a range of encounter rates and levels of hunger, we found that larval ladybird beetles showed intensive search behaviour for varying levels of encounter frequency, but did not alter their behaviour when hunger level was manipulated. We provide a conceptual model that explains why encounter frequency is more likely than hunger level to determine search behaviour. We discuss the relative merits of Fourier transforms in relation to fractal dimension and single-value measures.

Physical-based model of fish movement in fish extraction facilities

Abstract Fish collection or diversion facilities are structures designed to remove fish from a channel where they may be endangered from pumps, power plants, or irrigation systems. The Tracy Fish Collection Facility in the Central Valley of California (USA) collects endangered and economically important species before they can enter the Delta Mendota Canal. We describe the structure, sensitivity, and preliminary validation of a model that moves fish through this louver-type fish collection facility. The model is individual-based and moves fish subject to fundamental physical forces in the flowing medium and simple obstacle avoidance behaviors. Fluid dynamics are obtained by solving the Navier–Stokes equations. The primary model output is the salvage efficiency of the facility design. Monte Carlo simulation showed that the mean salvage efficiency is within the variability of field estimates. The most sensitive variables of the model are the initial cross-channel position of the fish and its initial energy reserves. The implications of our results for future collection facility designs are discussed.

Leading Students to Investigate Diffusion as a Model of Brine Shrimp Movement

Integrating experimental biology laboratory exercises with mathematical modeling can be an effective tool to enhance mathematical relevance for biologists and to emphasize biological realism for mathematicians. This paper describes a lab project designed for and tested in an undergraduate biomathematics course. In the lab, students follow and track the paths of individual brine shrimp confined in shallow salt water in a Petri dish. Students investigate the question, “Is the movement well characterized as a 2-dimensional random walk?” Through open, but directed discussions, students derive the corresponding partial differential equation, gain an understanding of the solution behavior, and model brine shrimp dispersal under the experimental conditions developed in class. Students use data they collect to estimate a diffusion coefficient, and perform additional experiments of their own design tracking shrimp migration for model validation. We present our teaching philosophy, lecture notes, instructional and lab procedures, and the results of our class-tested experiments so that others can implement this exercise in their classes. Our own experience has led us to appreciate the pedagogical value of allowing students and faculty to grapple with open-ended questions, imperfect data, and the various issues of modeling biological phenomena.

Assembly rules for Greater Antillean Anolis lizards

SummaryUsing previously published data, several models were constructed to predict the distribution of Anolis lizard species on a set of sites on Puerto Rico and Jamaica. The models form a series with increasing ecological detail. The simpler “null” models are based on randomly created species-site matrices using progressively greater dependency on the observed matrix. The remaining models hypothesize that competition is the most important biotic interaction determining the intra-island distribution of the lizards. “Simple” competition models test the predictive power of simple statistical descriptions relating intensity of competition and ecological variables such as niche overlap and body size ratios. More complicated models are based on the ecomorph model of Williams (1972) and use the lizard resourceuse data of several niche dimensions (e.g., perch diameter and height). These models are derived from Puerto Rican data and tested against Jamaican data. The primary statistical tool used to test the accuracy of these models in the kappa statistic (Fleiss 1973) which assesses the degree of agreement in a contingency table relative to that expected by chance. The model structure is based on generative grammars (Haefner 1981), but is also related to artificial intelligence expert systems. Model comparisons indicate the following. (1) Only those null models constrained by the marginals of the observed species-site matrix agree with observed data. (2) Simple competition models based on fixed size ratios and/or fixed levels of allowable overlap do not agree well. (3) A complex competition model developed for Puerto Rico also shows significant agreement with lizard distributions on Jamaica, but is not better than a constrained null model. (4) If allowance is made for the restricted distribution of A. sagrei, a recent colonist of Jamaica, agreement of the competition model increases dramatically. It is predicted that A. sagrei would persist following an experimental transplant to eastern Jamaica.

Ecosystem assembly grammars: Generative capacity and empirical adequacy

Abstract Using models based on generative grammars a theory of ecosystem assembly can be formulated that maps a set of species onto a set of environments (Haefner, 1977) . Such a theory must incorporate a minimal set of ecological properties in order to correctly describe the adaptive strategies of species and the non-random collection of species comprising an ecosystem. These properties include (1) concordance between activities performed by the individuals of a species, (2) the elaboration of niches due to species invasion, (3) concordance between resources and the users of resources, and (4) the plasticity of species behavior. These properties are used to define the criteria for the weak and strong empirical adequacy of grammars. Weak empirical adequacy of a grammar is the ability of a grammar to generate the sentences of a language. Strong empirical adequacy of a grammar is the ability of a grammar to generate the correct relationships between the elements of the sentences of a language. The adequacy of the members of the Chomsky hierarchy (regular grammars, context-free phrase-structure grammars, context-sensitive phrase-structure grammars, and transformational grammars) is evaluated by comparing their generative capacities and the criteria for empirical adequacy. This analysis indicates that for the representations of the phenomena considered strong empirical adequacy requires at least the generative capacity of a transformational grammar.

PARALLEL IMPLEMENTATIONS OF INDIVIDUAL-BASED MODELS IN BIOLOGY : BULLETIN-AND NON-BULLETIN-BOARD APPROACHES

Particle-based models are simulations in which the discrete representation of physical phenomenon involves interacting particles. This paper studies the efficiency of two different methods of implementing these models on a network of UNIX workstations. Two data parallel methods of modeling particles are tested: bulletin-board and non-bulletin-board. In the former method, the programs communicate through a logically shared, associative memory called a bulletin-board. The simulated particles are distributed among the workstations dynamically as the processing load on the processors changes. In the latter method, the particles are divided amongst the networked workstations statically at load time. The simulated system is a collection of ants moving and foraging in a two-dimensional space. This paper analyzes and compares the execution times of both implementations for different combinations of particles and number of workstation, using speed-up, tuple granularity and communication cost as measures. Analysis shows that the bulletin-board method is better for particle-based simulations when the correct granularity is chosen.

Carrying BioMath education in a Leaky Bucket.

In this paper, we describe a project-based mathematical lab implemented in our Applied Mathematics in Biology course. The Leaky Bucket Lab allows students to parameterize and test Torricelli’s law and develop and compare their own alternative models to describe the dynamics of water draining from perforated containers. In the context of this lab students build facility in a variety of applied biomathematical tools and gain confidence in applying these tools in data-driven environments. We survey analytic approaches developed by students to illustrate the creativity this encourages as well as prepare other instructors to scaffold the student learning experience. Pedagogical results based on classroom videography support the notion that the Biology-Applied Math Instructional Model, the teaching framework encompassing the lab, is effective in encouraging and maintaining high-level cognition among students. Research-based pedagogical approaches that support the lab are discussed.