Generalized population dynamics model of aphids in wheat based on catastrophe theory

Abstract

Wheat aphids are major wheat sap sucking pests found throughout the world. The analysis of wheat aphid population dynamics to develop aphid control strategies is therefore important. Even if all factors that control the size of aphid populations are known, several mathematical tools are needed to help us understand their combined effect. Based on the knowledge of population ecology and catastrophe theory, we proposed a generalized population dynamics model to describe variation of wheat aphid populations and obtained a dynamic threshold function for aphid control. Field survey data from 1997 to 2002 were used to validate this model. The results indicated the model could predict the results of practical measures against a pest if the factors of their immediate effects are known or could be estimated. By explaining and forecasting the size of an aphid outbreak and its probability of occurrence, this catastrophe model can provide a scientific basis for wheat aphid control.

Introduction

Aphids are major wheat pests found worldwide, causing significant yield losses in winter wheat (Emden and Harrington, 2007). There has been a marked increase in the amount of insecticide used to prevent yield losses. Analysis of the fluctuation of aphid population dynamics and establishment of mathematical models are necessary for the development of aphid control strategies involving less insecticide (Carter et al., 1982; Carter, 1985; Zhou and Carter, 1989). Aphid population dynamics is complicated because it is affected by numerous external factors, such as host plants, natural enemies, meteorology and insecticide (Watt, 1979; Chambers et al., 1986; Carter, 1987; Girma et al., 1990; Archer et al., 1998). Despite continuous changes in these factors, aphid population variation is not always continuous. Aphid numbers may initiate rapid development or decrease abruptly. After a sudden change in aphid populations, even if the external factors return to the state before the sudden change, the number of aphids does not immediately return to the original level unless external factors become more hospitable or inhospitable to living aphids. These two phenomena correspond to two flags (sudden transitions and hysteresis) of catastrophe theory (Zhao, 1991; Wu et al., 2014). Because both flags occur in the fluctuation of an aphid population, catastrophe models are used to analyze aphid population dynamics (Jones, 1977; Gilmore, 1993).

Catastrophe theory was first proposed by Thom to describe how the state of a system is discontinuously changed with continuous motion of control variables (Thom, 2018). Catastrophe theory can effectively link continuous smooth variation phenomena with sudden dispersion phenomena. Common examples include the sudden collapse of a bridge under slowly mounting pressure and the freezing of water when the temperature is gradually decreased. There are seven types of elementary catastrophe models, including fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic and parabolic umbilic catastrophe model. The first four models have one state variable, whereas the other three have two state variables. As the number of state variables and control variables increase, catastrophe models become increasingly complex. Catastrophe theory has been applied successfully in physical, social, and biological sciences (Zeeman, 1977; Poston and Stewart, 1978). Catastrophe theory is used to describe phenomena of sudden changes in continuous motion, which is related to the qualitative theory of the solution of differential equations (Gilmore, 1993). Thus, the differential equation model can be combined with the catastrophe theory. In the study of aphid population dynamics, some differential equation models have already been established by modifying the logistic model, which can be combined with fold (Zhao et al., 1988), cusp (Zhao et al., 2005), swallowtail (Piyaratne et al., 2013) and butterfly catastrophe models (Wu et al., 2014). All of these models can delineate the regularity of the growth and decline of an aphid population under the influence of different factors. However, it is difficult to determine what type of catastrophe model is a more appropriate choice for different actual data sets.

In this paper, we built a generalized population dynamics (GPD) model with three external factors (meteorological conditions, natural enemies and insecticide) to provide a more reasonable explanation for the choice of catastrophe models. The parameters of this model were estimated using three appropriate methods as explained in Section 2.4: (1) Catastrophe progression method. This comprehensive assessment method is used to obtain the composite index of meteorological factors. (2) Natural enemy unitization. This method illustrates the effective differences of the natural enemies. (3) The two-stage method. This method is used to solve parameter estimation of differential equations. Ultimately, field survey data over five years (1997–2001) were used to validate the GPD model, and the data of 2002 were used for prediction. In summary, our objectives are to (1) determine the main influential factors of aphid populations; (2) develop a generalized population dynamics (GPD) model for an aphid population; (3) demonstrate how to estimate parameter values; (4) select a suitable catastrophe model and apply the model to the field survey data; (5) provide a reference for farmers or policy makers for the prevention and control of aphids.