Functional Response Models Research Articles

Modeling the differential functional responses and selectivity of a marine copepod to nano/microplastics in mixture

Nano- and microplastics (NMPs) pollution is widespread in the oceans, posing potential risks to marine species. This study examined the accumulation capacity and selectivity potentials of NMPs by a marine copepod Parvocalanus crassirostris under different food mixtures by modeling the combined biokinetic and functional response. We investigated two sizes of NMPs (200 nm and 5 µm) across a concentration gradient (0 - 5000 µg/L) and varying diatom abundances (0, 104, 105 cells/mL). Fluorescence imaging and quantification revealed that P. crassirostris actively ingested NMPs at low concentration. Accumulation increased with NMPs concentration but eventually saturated due to gut capacity limits, following a Holling type II functional response (i.e., hyperbolic curve). Our novel functional response model estimated the key parameters and demonstrated that the maximum accumulation reached 5.3 % of dry weight with averaged half-saturation constants of 229 µg/L. The size of NMPs did not significantly affect the total accumulation or satiety levels. The presence of diatoms influenced the feeding selectivity and decreased the microplastic accumulation by 73 % at 105 cells/mL, while facilitating nanoplastic accumulation by 81 % at 104 cells/mL. This study enhanced our understanding of NMPs bioavailability and environmental fate in marine ecosystems.

Resilience mechanisms of Trichopria drosophilae (Hymenoptera: Diapriinae) under global extreme cooling: insights into parasitic response and physiological adaptation.

Global climate warming and frequent extreme low-temperature events have made it essential to investigate the impact of low temperatures on parasitic wasps to protect and strengthen farmland biodiversity, which in turn enhances the biological control potential of natural enemies such as parasitic wasps. We systematically examined how low-temperature stress affects the parasitic functional response of Trichopria drosophilae to Drosophila suzukii (Diptera: Drosophilidae) pupae. Our findings indicate that the parasitic behavior of T. drosophilae towards D. suzukii pupae aligns with the Holling II functional response model following exposure to different temperatures. Within the temperature range of 8 °C to -8 °C, lower temperatures correlated decreased instantaneous attack rate of T. drosophilae and an increase in processing time. The search constant Q initially increased and then decreased with declining temperatures. Short-term low-temperature stress negatively impacted the parasitic and searching abilities of T. drosophilae but did not alter its parasitic functional response model. Notably, short-term low-temperature stress had minimal effects on the water content, protein content, and total sugar content of male and female T. drosophilae adults. However, as temperatures decreased, the activities of key enzymes, including GAPDH, SOD, T-AOC, and malondialdehyde (MDA), exhibited an initial increase followed by a decrease. Conversely, the activities of LDH and HOAD decreased, while the activities of CAT and POD increased. Further study on the effect of short-term low temperature on T. drosophilae can provide a research basis for the large-scale production and low-temperature refrigeration technology of T. drosophilae, and provide a scientific basis for its efficient use in the field.

Functional responses correspond to stable isotope-based trophic positions among four invasive Ponto-Caspian mysid species (Crustacea, Mysida)

Determining the consumptive effects of non-native predators, for which several direct and indirect methods have been applied, is a common goal in invasion biology. Functional responses and stable isotope analysis are among the most widely used approaches; however, they are rarely used in combination. In this study, we used these two complementary methods to compare the predatory impacts of four invasive Ponto-Caspian mysids on zooplankton in a habitat (Danube River) where all four species coexist. The order of the species based on the attack rates of the functional response models matched with their mean δ15N values, indicating a correspondence between their predatory potentials and trophic positions in their invaded habitat. Hemimysis anomala had the highest attack rate on zooplankton presumably due to its higher degree of specialization linked to its pelagic lifestyle. Contrary to our expectations, the largest species, Paramysis lacustris, had an intermediate predatory efficiency and trophic position, similar to those of Limnomysis benedeni but higher than those of the smallest species, Katamysis warpachowskyi. Nevertheless, all of the four species exhibited a considerable predatory potential, suggesting that any of them can contribute significantly to their combined predatory impact depending on their densities. The congruence between the results of the two methods shows that the species can realize their different predatory potentials in their invaded environment, indicating favorable conditions (i.e., food availability and spatial heterogeneity) which allowed dietary differentiation. We recommend the combined use of functional responses and stable isotope analysis, which might allow robust conclusions to be drawn on the trophic ecology of the species and also provide further insights into the studied ecosystem.

A doubly robust estimator for the Mann Whitney Wilcoxon rank sum test when applied for causal inference in observational studies

ABSTRACT The Mann–Whitney–Wilcoxon rank sum test (MWWRST) is a widely used method for comparing two treatment groups in randomized control trials, particularly when dealing with highly skewed data. However, when applied to observational study data, the MWWRST often yields invalid results for causal inference. To address this limitation, Wu et al. (Causal inference for Mann–Whitney–Wilcoxon rank sum and other nonparametric statistics, Stat. Med. 33 (2014), pp. 1261–1271) introduced an approach that incorporates inverse probability weighting (IPW) into this rank-based statistic to mitigate confounding effects. Subsequently, Mao (On causal estimation using U-statistics, Biometrika 105 (2018), pp. 215–220), Zhang et al. (Estimating Mann Whitney-type causal effects, J. Causal Inference 7 (2019), ARTICLE ID 20180010), and Ai et al. (A Mann–Whitney test of distributional effects in a multivalued treatment, J. Stat. Plan. Inference 209 (2020), pp. 85–100) extended this IPW estimator to develop doubly robust estimators. Nevertheless, each of these approaches has notable limitations. Mao's method imposes stringent assumptions that may not align with real-world study data. Zhang et al.'s (Estimating Mann Whitney-type causal effects, J. Causal Inference 7 (2019), ARTICLE ID 20180010) estimators rely on bootstrap inference, which suffers from computational inefficiency and lacks known asymptotic properties. Meanwhile, Ai et al. (A Mann–Whitney test of distributional effects in a multivalued treatment, J. Stat. Plan. Inference 209 (2020), pp. 85–100) primarily focus on testing the null hypothesis of equal distributions between two groups, which is a more stringent assumption that may not be well-suited to the primary practical application of MWWRST. In this paper, we aim to address these limitations by leveraging functional response models (FRM) to develop doubly robust estimators. We demonstrate the performance of our proposed approach using both simulated and real study data.

Field and laboratory microplastics uptake by a freshwater shrimp.

Microplastics are widespread pollutants, but few studies have linked field prevalence in organisms to laboratory uptakes. Aquatic filter feeders may be particularly susceptible to microplastic uptake, with the potential for trophic transfer to higher levels, including humans. Here, we surveyed microplastics from a model freshwater shrimp, common caraidina (Caridina nilotica) inhabiting the Crocodile River in South Africa to better understand microplastic uptake rates per individual. We then use functional response analysis (feeding rate as a function of resource density) to quantify uptake rates by shrimps in the laboratory. We found that microplastics were widespread in C. nilotica, with no significant differences in microplastic abundances among sampled sites under varying land uses, with an average abundance of 6.2 particles per individual. The vast majority of microplastics found was fibres (86.1%). Shrimp microplastic accumulation patterns were slightly higher in the laboratory than the field, where shrimp exhibited a hyperbolic Type II functional response model under varying exposure concentrations. Maximum feeding rates of 20 particles were found over a 6 h feeding period, and uptake evidenced at even the lowest laboratory concentrations (~10 particles per mL). These results highlight that microplastic uptake is widespread in field populations and partly density dependent, with field concentrations corroborating uptake rates recorded in the laboratory. Further research is required to elucidate trophic transfer from these taxa and to understand potential physiological impacts.

Effects of flow speed and prey density on the rate and efficiency of prey capture in zooplanktivorous coral-reef fishes

Holling’s classical functional response model describes the mechanistic foundations of the relationships between predation rate and prey density. As such, the model is pertinent to predators that actively search for prey, but not to stationary predators in which additional factors, such as flow speed, determine the rates of prey encounter. The main objective of this study was to measure the rates and corresponding efficiencies of zooplanktivory among different common species of coral-reef fishes under a wide range of prey densities and current speeds. All our experiments were carried out in a flume with different combinations of flow speeds (3-28.5 cm/s) and prey densities (210 - 1050 prey m-3). Nauplii of Artemia salina were used as prey. Despite major differences in the taxonomic origin of the studied species, their morphologies, and the types of shelters they use, the foraging performances of the fish, their predation rates, and the way those rates were affected by prey density and flow speed were surprisingly similar. Under a fixed prey density, capture rates did not change much as function of flow speed. Under conditions of equal prey flux, predation rates were always higher under conditions of high density and weaker flow than under lower density and faster flow. A sharp decline in capture efficiency with increasing flow speed was explained by a corresponding narrowing of the fish’s body orientation relative to the flow. In other words, with increasing flow speed, the fish gradually became more-narrowly oriented head-on onto the flow, exhibiting a decrease in the frequency of body turns (“maneuverability”). These trends, especially the reduced maneuverability under strong currents, can explain our findings that predation rates did not increase when the flow, hence prey flux, increased. Inter-specific differences in predation rates and efficiencies, however small, agree well with observed differences in the type of habitats the different species occupy.

Corrigendum: Geometric complexity and the information-theoretic comparison of functional-response models

Corrigendum: Geometric complexity and the information-theoretic comparison of functional-response models

Effect of egg production dynamics on the functional response of two parasitoids.

Functional response describes the number of hosts attacked by a parasitoid in relation to host densities and plays an important role by connecting behavioral-level processes with community-level processes. Most functional response studies were carried out using simple experimental designs where the insects were confined to a plain and small arena with different host densities during a fixed period of time. With these designs, other factors that might affect the functional response of parasitoids were not analyzed, such as fecundity, age, and experience. We proposed a series of latent-variables Markovian models that comprised an integrated approach of functional response and egg production models to estimate the realized lifetime reproductive success of parasitoids. As a case study, we used the parasitoids Anagyrus cachamai and A. lapachosus (Hymenoptera: Encyrtidae), two candidate agents for neoclassical biocontrol of the Puerto Rican cactus pest mealybug, Hypogeococcus sp. (Hemiptera: Pseudococcidae). The tested species were assessed according to their physiology and prior experience. We estimated the number of mature eggs after emergence, egg production on the first day, egg production rate, the proportion of eggs resorbed, egg resorption threshold, and egg storage capacity. Anagyrus cachamai and A. lapachosus both presented a type III functional response. However, the two parasitoids behaved differently; for A. cachamai, the number of parasitized hosts decreased with female age and depended on the number of mature eggs that were available for oviposition, whereas A. lapachosus host parasitism increased with female age and was modulated by its daily egg load and previous experience. The methodology presented may have large applicability in pest control, invasive species management, and conservation biology, as it has the potential to increase our understanding of the reproductive biology of a wide variety of species, ultimately leading to improved management strategies.

Partial exploitation of host egg patches resulting from a high rejection rate of healthy hosts cautions the mechanistic use of functional response models

Abstract The functional response of parasitoids is a main component of models that describe host–parasitoid interactions. When the functional response assumed in a model (e.g., a population model) is inaccurate, predictions by the model based on the functional response also become unreliable. This study reports that conventional functional response models may commonly fail to capture the actual functional response of parasitoids and discuss its implications. To assist the discussion, an experiment was conducted to describe interactions between the pupal parasitoid Dirhinus giffardii (Hymenoptera: Chalcididae) and its host Bactrocera dorsalis (Diptera: Tephritidae) in the laboratory. Results showed that the parasitoids typically interacted with all hosts in the environment but parasitised a small fraction of them (e.g., most encountered hosts were rejected), which deviates from the assumption of mechanistic functional response models. An increasing number of published studies on functional response also suggest (although circumstantially) the rejection of unparasitised hosts is common, suggesting that this behaviour is widespread among parasitoid species. By using the experimental results, the study discusses why existing hypotheses are insufficient for the observed behaviour and the implications of this behaviour within the realm of theories surrounding host–parasitoid interactions.

Estimation of predation rate and handling time of boll weevil larvae by Marava arachidis (Dermaptera: Labiidae) using different mathematical methods.

Anthonomus grandis grandis (Coleoptera: Curculionidae) is a pest with a large potential for destruction in cotton crops, causing damage to the cotton reproductive structures. The earwig Marava arachidis (Dermaptera: Labiidae), is an important reference as a predator in several crops and being easy to rear in the laboratory. To analyze the potential biocontrol of M. arachidis of A. grandis grandis larvae, a study of predatory capacity was conducted using a functional response model. A. grandis grandis larvae were exposed to the predator at densities 1, 2, 4, 6, and 8 larvae (= prey/predator / Petri dish), with 30 replications at each density. Contact between the predator and the prey occurred for 24 hours; after this period, the level of predation of M. arachidis was assessed based on the proportion of preyed larvae. The linear logistic regression coefficient was used with a beta-binomial generalized linear model to determine the functional response. The negative signal of the linear coefficient and the goodness-of-fit tests revealed a quadratic or type II functional response, with the number of prey varying from 1.00 larva (density of 1 larva/predator) to 6.50 larvae (density of 8 larvae/predator). Therefore, the results of the present study demonstrate a high predatory capacity of M. arachidis on A. grandis grandis larvae.

BeatProfiler: Multimodal In Vitro Analysis of Cardiac Function Enables Machine Learning Classification of Diseases and Drugs.

Goal: Contractile response and calcium handling are central to understanding cardiac function and physiology, yet existing methods of analysis to quantify these metrics are often time-consuming, prone to mistakes, or require specialized equipment/license. We developed BeatProfiler, a suite of cardiac analysis tools designed to quantify contractile function, calcium handling, and force generation for multiple in vitro cardiac models and apply downstream machine learning methods for deep phenotyping and classification. Methods: We first validate BeatProfiler's accuracy, robustness, and speed by benchmarking against existing tools with a fixed dataset. We further confirm its ability to robustly characterize disease and dose-dependent drug response. We then demonstrate that the data acquired by our automatic acquisition pipeline can be further harnessed for machine learning (ML) analysis to phenotype a disease model of restrictive cardiomyopathy and profile cardioactive drug functional response. To accurately classify between these biological signals, we apply feature-based ML and deep learning models (temporal convolutional-bidirectional long short-term memory model or TCN-BiLSTM). Results: Benchmarking against existing tools revealed that BeatProfiler detected and analyzed contraction and calcium signals better than existing tools through improved sensitivity in low signal data, reduction in false positives, and analysis speed increase by 7 to 50-fold. Of signals accurately detected by published methods (PMs), BeatProfiler's extracted features showed high correlations to PMs, confirming that it is reliable and consistent with PMs. The features extracted by BeatProfiler classified restrictive cardiomyopathy cardiomyocytes from isogenic healthy controls with 98% accuracy and identified relax90 as a top distinguishing feature in congruence with previous findings. We also show that our TCN-BiLSTM model was able to classify drug-free control and 4 cardiac drugs with different mechanisms of action at 96% accuracy. We further apply Grad-CAM on our convolution-based models to identify signature regions of perturbations by these drugs in calcium signals. Conclusions: We anticipate that the capabilities of BeatProfiler will help advance in vitro studies in cardiac biology through rapid phenotyping, revealing mechanisms underlying cardiac health and disease, and enabling objective classification of cardiac disease and responses to drugs.

Understanding the hydra effect in predator-dependent functional response models

Understanding the hydra effect in predator-dependent functional response models

Hybrid modeling for grassland productivity prediction: A parametric and machine learning technique for grazing management with applicability to digital twin decision systems

CONTEXTMonitoring Aboveground Net Primary Production (ANPP) is critical to assess not only the current ecosystem status but also its long-term dynamics. In rangelands, the seasonal dynamics of ANPP determines forage availability, stock density, and livestock productivity. OBJECTIVETo develop a hybrid model to be used as a prediction engine for ANPP in the native grasslands of Uruguay. The model combines a parametric component based on the seasonal dynamics of ANPP, and an artificial neural network (ANN) component used to model the remaining non-linearities, which are mainly related to precipitation and temperature variability. The output of hybrid model is proposed as the “virtual entity” of a digital twin support decision system where the “physical entity” is characterized by a collection of bi-weekly (fortnight) ANPP estimates. METHODSFortnight ANPP data were calculated from MODIS EVI for the 2001–2020 period. A sigmoidal functional response, having three parameters with an explicit biological interpretation, was fitted to the accumulated ANPP as a function of time. Forecasts were generated by extrapolating the sigmoidal functional response fit up to four fortnights ahead. From these fits, we obtained the fortnight ANPP values by differentiating the accumulated fortnight ANPP. Predictions (up to four fortnights) were generated for each fortnight and year. The residuals from these fits were modeled using a multilayer perceptron trained by backpropagation using climate variables as independent variables. RESULTS AND CONCLUSIONSThe sigmoidal functional response model fit was highly significant for the accumulated ANPP profile. This model also had a high explanatory power for the accumulated ANPP curve. The median of the percentage absolute residuals for forecasts made 1 to 4 fortnights ahead ranged from 17% to 18%. The ANN significantly reduced this unexplained variability in ANPP, showing a median reduction in residuals of 35%, 31%, 30%, and 30% for 1 to 4 fortnights ahead forecasts, respectively, when compared to predictions from the sigmoidal functional response fit. SIGNIFICANCEBy integrating both parametric and machine learning techniques, the hybrid model developed can make accurate predictions in a way that is both efficient and dependable. The hybrid model not only represents an advantage in terms of predictive power, but it also allows for a deeper understanding of the basic ecological processes involved in forage production.

Brown bear (Ursus arctos) foraging in a mosaic of spatially discrete and variable habitats over 25 years of shifting Pacific salmon densities

Many foraging models assume “perfect information” and “free movement” when describing predator foraging behavior, although this is rare in nature. Here, we quantified predation by brown bears ( Ursus arctos Linnaeus, 1758) on adult sockeye salmon ( Oncorhynchus nerka (Walbaum, 1792)) in a series of spatially proximate ponds that largely satisfied both assumptions. Salmon abundance varied among years, but pond area and depth were fixed, allowing us to examine interactions between prey abundance and habitat features. We applied versions of two models to 25 years of data on the number and proportion of salmon killed by bears, modifying these models to include habitat features and temporal variability. The functional response model with a year effect fit the data well, indicating bears could take almost all salmon in ponds when salmon were scarce, but bears were sated when salmon were abundant. The proportion of salmon killed by bears was similar across habitats after correcting for pond depth and area. Overall, bears foraged across all habitats but killed higher proportions of salmon in smaller and shallower habitats, consistent with ease of capture.

Bacteria as ecosystem engineers: Unraveling clues through a novel functional response and tritrophic model

Bacteria play a crucial role in regulating the nutrient cycle of ecosystems, making their abundance essential for the sustainability of these environments. Maintaining a thriving bacterial population is achieved through complex nutrient–bacteria–protozoa interactive dynamics, which can be likened to a tritrophic food chain model. The interaction between bacteria and protozoa is governed by a critical factor known as the bacteria–protozoa functional response, which serves as the foundation for developing this tritrophic model. However, existing functional responses have shown limitations in accurately describing the intricacies of the bacteria–protozoa interaction. One significant drawback is the neglect of bacterial behavioral traits. To address this issue, we consider the concept of cooperation as a group defense mechanism employed by bacteria facilitated through a quorum-sensing communication process. By incorporating the cooperation trait into the functional response, our model offers a more comprehensive understanding of the complex tritrophic food chain dynamics. We evaluate the stability of different equilibrium points, along with Hopf-bifurcation around the coexistence equilibrium point. We find that a balance between strong group defense and moderate cooperation is essential for bacteria sustainability and overall system stability. Our results also elaborately address the effects of the increasing group defense through the bistable equilibrium followed by a branch point and saddle–node bifurcation. Through comprehensive analyses and simulations, we examine the paradox of enrichment in nutrition flow at the community level and explore how nutrient washout controls system stability. This innovation not only enhances our comprehension of ecosystem sustainability but also opens up new avenues for studying the intricate relationships that govern the overall balance of nature.

Kernel mean embedding of probability measures and its applications to functional data analysis

AbstractThis study intends to introduce kernel mean embedding of probability measures over infinite‐dimensional separable Hilbert spaces induced by functional response statistical models. The embedded function represents the concentration of probability measures in small open neighborhoods, which identifies a pseudo‐likelihood and fosters a rich framework for statistical inference. Utilizing Maximum Mean Discrepancy, we devise new tests in functional response models. The performance of new derived tests is evaluated against competitors in three major problems in functional data analysis including Function‐on‐Scalar regression, functional one‐way ANOVA, and equality of covariance operators.

KAPASITAS POTENSI PARASITISME Trichogramma sp. (Hymenoptera: Trichogrammatidae) TERHADAP Corcyra sp.

Parasitism research Trichogramma sp. (Hymenoptera: Trichogrammatidae) has been carried out with the aim of studying the functional response of the parasitoid Trichogramma sp. Functional response is one measure to determine the effectiveness of a predator or parasitoid as an agent in biological control, so that it can express a change in the number of preys attacked by individual predators. Temperature is one of the most important host-parasitoid interactions in functional responses. The method used in the parasitoid parasitism Trichogramma sp. on the host Corcyra sp. using four levels of egg density Corcyra sp. with two different temperature treatments (27OC and room temperature). The results show the functional response model of the parasitoid Trichogramma sp. against its host Corcyra sp. described in the type I model. Parasitization level of Trichogramma sp. showed the same results for both temperature treatments, however, it also shows differences in the speed of predation. It also showed linearity between the parasitoids Trichogramma sp. with the host Corcyra sp. Parasitism rates are still increasing, along with an increase in host density. This research can be a starting point to improve the control of Corcyra sp. using the parasitoid Trichogramma sp. in the field.

FUNCTIONAL RESPONSE OF PREDATOR Paederus sp. (COLEOPTERA: STAPHYLINIDAE)

Research on predatory predation of Paederus sp. (Coleoptera: Staphylinidae) was carried out to study the functional response of the predator Paederus sp. Functional response is a form and size predictor of consumer impact on resource populations, determining the effectiveness of a predator as an agent in biological control. Predation time in hungry and not-hungry conditions influences predator-prey interactions, which are essential in functional response. The method used in predating Paederus sp. on the prey of aphids was Aphid spp. They are using four levels of prey density with two different predator treatments. The results show the functional response model of Paederus sp. predators. against the prey of Aphid spp. Described in a type II model. The level of predation of Paederus sp. showed different results for the two treatments and an increasing cystoid curve at a rising rate. The level of predation is still increasing, along with the increase in host density, and can be a starting point for controlling the Aphid spp and using predators Paederus sp. in the field.

Qualitative analysis and phase of chaos control of the predator-prey model with Holling type-III

In this study, we investigate the dynamics of a discrete-time with predator-prey system with a Holling-III type functional response model. The center manifold theorem and bifurcation theory are used to create existence conditions for flip bifurcations and Neimark-Sacker bifurcations. Bifurcation diagrams, maximum Lyapunov exponents, and phase portraits are examples of numerical simulations that not only show the soundness of theoretical analysis but also show complicated dynamical behaviors and biological processes. From the point of view of biology, this implies that the tiny integral step size can steady the system into locally stable coexistence. Yet, the large integral step size may lead to instability in the system, producing more intricate and richer dynamics. This also means that when the intrinsic death rate of the predator is high, this leads to a chaotic growth rate of the prey. The model has bifurcation features that are similar to those seen in logistic models. In addition, there is a bidirectional Neimark-Sacker bifurcation for both prey and predator, and therefore we obtain a direct correlation in symbiosis. This means that the higher the growth rate of the prey, the greater the growth rate of the predator. Therefore, the operation of predation has increased. The opposite is also true. Finally, the OGY approach is used to control chaos in the predator and prey model. which led to a new concept which we call bifurcation phase of control chaos.

A new model formulation for host depletion in parasitoids

A new model of host depletion in the functional response of parasitoids is presented. Most existing models describe the dynamics of prey consumption as a sequential-memoryless/Markovian process of searching for, handling, and consuming prey, but the feeding interaction between a parasitoid and its host is different. One of the most important characteristics is that the host is not consumed, and remains in the arena. On the other hand, functional response models of solitary parasitoids assume that each host can accommodate an infinite number of oviposition events without being killed or becoming unsuitable as a host without the parasitoid noticing it. As a consequence, host depletion is never considered in parasitoid functional response models. To solve that issue, we extended the traditional random predator and random parasitoid functional response models of Rogers into a compartmental model formulated with ordinary differential equations, to group hosts with a different number of attacks to simulate how the parasitoid spreads the attacks between already parasitized and non-parasitized hosts. This model was further modified to simulate non-sequential memoryless search, handling and oviposition. We studied the behaviour of these models to understand how a variation of their parameters led to different shapes of functional response curves and statistical distributions of the number of parasitoid eggs laid per host. The results showed that the expected curves differ from those predicted by the simpler Holling's disk equation (type II functional response), random parasitoid models (no depletion), and the random predator model (proportional depletion). Finally, the non-sequential-memoryless models predict straight curves resembling a type I functional response at low host densities.