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This paper presents and compares four mathematical models with unique spatial effects for a prey-predator system, with
*Tetranychus urticae* as prey and
*Phytoseiulus persimilis* as predator.
*Tetranychus urticae*, also known as two-spotted spider mite, is a harmful plant-feeding pest that causes damage to over 300 species of plants. Its predator,
*Phytoseiulus persimilis*, a mite in the Family Phytoseiidae, effectively controls spider mite populations. In this study, we compared four mathematical models using a numerical simulation. These models include two known models: self-diffusion, and cross-diffusion, and two new models: chemotaxis effect model, and integro diffusion model, all with a Beddington-De Angelis functional response. The modeling results were validated by fitting experimental data. Results demonstrate that interaction scheme plays an important role in the prey-predator system and that the cross-diffusion model fits the real system best. The main contribution of this paper is in the two new models developed, as well as the validation of all the models using experimental data.

The two-spotted spider mite, Tetranychus urticae, is a species of plant-feeding mites generally considered to be a pest. It is the most widely known member of the Family Tetranychidae and is a harmful pest in greenhouses and on field-grown crops. Previous reports have stated that Tetranychus urticae infests over 300 species of plants, including ornamental plants such as arborvitae, azalea, camellia, citrus, evergreens, hollies, ligustrum, pittosporum, pyracantha, rose, and viburnum; fruit crops such as blackberries, blueberries, and strawberries; and vegetable crops such as tomatoes, squash, eggplant, and cucumber [

Insects have three pairs of legs and three body regions (head, thorax, abdomen), but throughout most life stages, spider mites have four pairs of legs and one body region. Tetranychus urticae is distinguishable by two large dark green spots on the dorsal area of the abdomen. Depending on the host plant and other environmental factors, such as temperature and light, the color of Tetranychus urticae varies from light green, dark green, brown, black, and orange [

Predators beneficially regulate spider mite populations. Five species of spider mite predators are commercially available in the United States for crop protection: Phytoseiulus persimilis, Mesoseiulus longipes, Neoseiulus californicus, Galendromus occidentalis, and Amblyseius fallicus. Predatory mites are distinguishable from spider mites due to longer legs, a more active life, and a faster pace of movement. Predators are often red or orange in color [

Previous work has attempted to determine biological mechanisms, including dispersal, underlying mechanism of the spider mite-Phytoseiulus persimilis interaction. When diffusion is introduced into a prey-predator system, both species attain uniform distributions in the domain after certain time. Diffusion acts as a stabilizer in a reaction-diffusion system [

Similar to predator interference and relative diffusion, another factor, called prey- taxis, introduce instability into this domain, and leading to the formation of spatial patterns. In the Lotka-Volterra logistic prey-predator model with prey-taxis, Sapoukhina et al. [

Phytoseiulus persimilis responds to odors released from leaves infested by Tetranychus urticae. Sabelis and Weel [

This paper presents and compares four prey-predator models with distinctive spatial effects as they apply to a two-spotted spider mite and Phytoseiulus persimilis system. The paper is organized as follows. Section 2 presents the four models, i.e. self-diffusion model, cross-diffusion model, chemotaxis effect model, and integro-diffusion model for a Tetranychus urticae and Phytoseiulus persimilis prey-predator dynamic system. Section 3 presents simulation results, and Section 4 compares experimental data with simulated data from various models. Section 5 discusses numerical simulation and model validation.

The dynamic relationship between predator and prey is a central ecological matter and a primary concern when modeling prey-predator interactions. A significant component of the prey-predator relationship is the functional response, which indicates the average number of prey killed per predator per unit of time. Two types of functional response are common: prey-dependent and predator-dependent. Prey-dependent implies that the functional response depends only on prey density; in a predator-dependent response, the function of response depends on both prey and predator densities. In the literature, the prey-dependent function has served as the basis for predator-prey theory, such as Holling’s Type II functional response [

by Beddington [

This paper employs the Beddington-DeAngelis response function [

where

r: intrinsic growth of spider mites

K: carrying capacity of spider mites

e: conversion rate of prey to predator

d: death rate of Phytoseiulus persimilis

a: maximum consumption rate

b: saturation constant

c: factor to scale the impact of predator interference.

Let

After manipulation, the following polynomial form is obtained:

This dynamic relationship is demonstrated in

According to

This paper fits the dynamic system (2) into four models: self-diffusion model and cross-diffusion model which are based on existing formulations, and chemotaxis effect model and integro diffusion model which are part of the contribution of this paper. These models are presented in the following sections.

The tendency for a species to move in the direction of lower species density is called

self-diffusion [

where:

d_{1}: self-diffusion coefficient for prey

d_{2}: self-diffusion coefficient for predator

For the Beddington-DeAngelis response function and logistic growth function, the corresponding polynomial form becomes:

Although the self-diffusion model demonstrates that the movement within a given species is independent of other species, prey may recognize predators and respond by moving away to avoid capture by predators in predator-prey systems. However, if predators recognize prey, this recognition may affect the rate or direction of their movement, thereby helping the predators find prey. This phenomenon, known as cross-diffusion, has recently received significant attention, as described in [

The general form of a cross-diffusion model for prey-predator interactions is presented as follows:

where:

d_{11} and d_{22}: self-diffusion coefficients of prey and predator, respectively

d_{12}and d_{21}: cross diffusion coefficients of predator and prey, respectively.

If d_{12} > 0 and d_{21} < 0, then the prey species tends to diffuse in the direction of lower concentration of the predator species and the predator species tends to diffuse in the direction of higher concentration of the prey species. Using the Beddington-DeAngelis response function and logistic growth function, the corresponding polynomial form becomes:

A large number of insects, animals, and humans rely on smell to convey information between species members. Predatory mites respond to volatile chemicals released by plants infested with spider mites, as shown in experiments using Y-tube olfactometers and chemical analyses [

The predator-prey model with chemotaxis effect can be written as:

where:

d_{3}: effect of social behavior

T: sensitivity coefficient of predators to heterogeneous density distribution of prey

Integro-differential equations (IDEs) share continuous-space and continuous population assumptions of partial-differential equation (PDE) models. The PDE model focuses on localized movement (diffusion) of individuals, while IDE models focus on long- range movement. In this case, both prey and predator can move a long distance.

The predator-prey model with IDE can be written as:

Equation (8) considers movement from all points in space D (labeled y in the integral) to point x. T is the dispersal time of each species, and μ is a species parameter describing the diffusivity, or rate of dispersal, of each population. Movement rate, assumed to vary with distance, is described by kernel function K_{1} and K_{2}. Kernel function defines how movement rate decreases with distance, thus offering greater flexibility than the PDE model. Therefore, predation can occur over a variety of scales instead of being a local event.

In this section, dynamic simulations are performed with the four discussed models. The simulation is performed on a two-dimensional lattice with 100 × 100 cells. Spacing between each lattice cell was 1.25 unit distance, and the timing step was 0.05. Laplacian diffusion was calculated using finite difference, and Neumann boundary conditions were employed. The parameter used [_{11} (d_{1}) = 0.01, d_{22} (d_{2}) = 1, d_{12} = 0.005, and d_{21} = −0.001. Additional parameters for the chemotaxis effect model were d_{3} = 0.005, T = 0.01, for integro diffusion model were μ = 0.051, T = 2.5.

Prey and predator densities were compared at fixed locations of (50, 50) and (90, 90) within a 100 × 100 grid. The simulation ran 10,000 iterations with initial prey density 0.5 and predator density 0.2. Results of the self-diffusion model, cross-diffusion model, chemotaxis effect model, and integro diffusion model are shown in Figures 3-6, respectively. In all four figures, the left subfigure (a) represents prey and predator densities at the location (50, 50) and the right subfigure (b) represents these densities at the location (90, 90).

Simulation results in Figures 3-6 show that the chemotaxis effect model differs significantly from the other three models. The chemotaxis effect system did not achieve steady state by 10,000 iterations, while the other three dynamic systems achieved steady state at approximately 4000 - 6000 iterations. The reason for this could be predatory

mites move faster toward the higher density of prey area when attraction odors are present for predatory mites, thereby weakening system stability.

This section presents simulated patterns of formation among models. Using stable state as the initial condition, simulations were run with 0, 100,000, and 20,000 iterations. Corresponding results are shown in subfigure (a), subfigure (b), and subfigure (c) of Figures 7-10, respectively. Simulated results for self-diffusion, cross-diffusion, chemotaxis effect, and integro diffusion models are shown in Figures 7-10, respectively.

From Figures 7-10, it could be concluded that different spatial effect played important role towards the dynamic system. Using the chemotaxis effect model shows a larger range of density distributions of prey and predator than that of the other three models when they all began from the same steady state. For instance, after 20,000 iterations, the difference of density distribution of prey is around 0.45 for chemotaxis effect model while that for other models is around 0.0045, and the density distribution difference of

predator is about 0.05 for chemotaxis effect model while that of predator for other models is about 0.0005. This indicates that chemotaxis introduces more instability into the model. On the other side, the pattern for the integro diffusion model differed significantly from the other models, which is consistent with the model assumption that prey and predator system has long-range interaction during their movement. The simulation results verified the assumption of different spatial effect models and confirmed that different interaction scheme plays an important role in this prey-predator system.

This experiment, conducted by the entomology department at Kansas State University, was carried out on 24 individually-potted lima beans plants set in 8 × 3 arrays, with Phytoseiulus persimilis as predator and Tetranychus urticae as prey. The experiment lasted four weeks, and the total number of two-spotted spider mites and predator were counted every six days.

This section compares the number of two-spotted spider mites and its predator using a simulated model with the experimental (actual) observations. Parameters in Equation (1) for simulation were α = 20, b = 105, c = 45, d = 0.3, e = 0.25, r = 0.38, K = 800.

Simulation results are shown in Figures 11-14, where the dotted curve represents actual data from the experiment and the solid curve represents the number of prey and predator calculated from the simulated models. The number of prey comparison is presented on the subfigure (a) while the number of predator comparison is presented in the subfigure (b) of each figure.

From

Times/Days | Observations | |
---|---|---|

Number of prey | Number of predator | |

0 | 64 | 6 |

6 | 458 | 6 |

12 | 490 | 13 |

18 | 2238 | 67 |

24 | 1954 | 239 |

Time/Days | Different Models | ||||||
---|---|---|---|---|---|---|---|

Observations of Prey | Self-Diffusion | Cross-Diffusion | Chemotaxis Effect | Integro Diffusion | |||

0 | 64 | 64 | 64 | 64 | 64 | ||

6 | 458 | 266 | 267 | 268 | 301 | ||

12 | 490 | 1016 | 990 | 1285 | 1543 | ||

18 | 2238 | 2116 | 2107 | 2134 | 1761 | ||

24 | 1954 | 2171 | 1774 | 1802 | 1517 | ||

RMSE | 306.59 | 289.83 | 418.93 | 622.49 | |||

Time/Days | Different Models | ||||
---|---|---|---|---|---|

Observations of Predator | Self-Diffusion | Cross-Diffusion | Chemotaxis Effect | Integro Diffusion | |

0 | 6 | 6 | 6 | 6 | 6 |

6 | 6 | 8 | 7 | 5 | 4 |

12 | 13 | 11 | 11 | 48 | 93 |

18 | 67 | 175 | 152 | 412 | 433 |

24 | 239 | 417 | 386 | 588 | 488 |

RMSE | 104.16 | 84.85 | 245.97 | 224.72 |

Results show that the cross-diffusion model fits the two-spotted spider mite system best, with the smallest RMSE compared to the other three models for prey and predator number prediction. The integro diffusion model had the largest RMSE for prey number prediction while the chemotaxis effect model had the largest RMSE for predator number prediction.

This paper presented and analyzed four mathematical models with the Beddington- DeAngelis functional response [

Another conclusion that could be made is that, the two proposed models were theoretically reasonable. According to the simulation, the chemotaxis effect model was not as stable as the other three models, affirming that predator mites move faster and further when presented with attracting odors, thereby reducing system stability. The chemotaxis effect model lack of stability was also derived from the pattern formation simulation result. The result shows the range of density distribution of the chemotaxis effect model was much larger than that of the other three models when all models began from an identical steady state. On the other hand, the pattern for the integro diffusion model differed from the other models, which is consistent with the model assumption that prey and predator has long-range interaction during their movement.

In the validation process, results showed that all four models have good fit with the real system, with the cross-diffusion model having the best fit. For a future research, we plan to develop an agent-based model [

Kuang, Y., Ben- Arieh, D., Zhao, S.N., Wu, C.-H., Margolies, D. and Nechols, J. (2017) Mathematical Model for Two-Spotted Spider Mites System: Verification and Validation. Open Jour- nal of Modelling and Simulation, 5, 13-31. http://dx.doi.org/10.4236/ojmsi.2017.51002