The control of insect pests in agriculture is essential for food security. Chemical controls typically damage the environment and harm beneficial insects such as pollinators, so it is advantageous to identify targetted biological controls. Since predators are often generalists, pathogens or parasitoids are more likely to serve the purpose. Here, we model a fungal pathogen of aphids as a potential means to control of these important pests in cereal crops. Typical plant herbivore pathogen models are set up on two trophic levels, with dynamic variables the plant biomass and the uninfected and infected herbivore populations. Our model is unusual in that (i) it has to be set up on three trophic levels to take account of fungal spores in the environment, but (ii) the aphid feeding mechanism leads to the plant biomass equation becoming uncoupled from the system. The dynamical variables are therefore the uninfected and infected aphid population and the environmental fungal concentration. We carry out an analysis of the dynamics of the system. Assuming that the aphid population can survive in the absence of disease, the fungus can only persist (and control is only possible) if (i) the host grows sufficiently strongly in the absence of infection, and (ii) the pathogen transmission parameters are sufficiently large. If it does persist the fungus does not drive the aphid population to extinction, but controls it below its disease-free steady state value, either at a new coexistence steady state or through oscillations. Whether this control is sufficient for agricultural purposes will depend on the detailed parameter values for the system.

#### Keywords

- three-way interactions
- pest control
- mathematical model
- pathogenic fungi
- aphids
- crops

#### MSC 2010

- AMS 2010 MSC primary: 92D40
- secondary: 92D25
- 92D30

Aphids represent a very relevant pest in agriculture because they weaken and kill plants. To this end, they attach themselves to the most tender parts, where they are able to perforate the plant surface and then feed on the plant sap. In this way the plant receives less nutrient and suffers consequently. Aphids may be treated with pesticides, but this has negative ecological consequences; moreover, in recent times, aphids have undergone evolutionary changes in order to adapt to and to resist these human antagonistic practices, [5]. The need of finding alternative means for better fighting them arises naturally. Biological control could be a viable solution, [7]. It can be performed via several approaches.

For instance, specialised parasitoids, such as certain wasp species, can be used [6]. The latter lay their eggs inside the aphid body through their skin. After hatching, the wasp larva feeds on the aphid and finally emerges to the adult stage, by which action the aphid is killed. The latter however are subject to the counteraction of common facultative bacterial symbionts, that confer resistance to these natural enemies, reducing the rate at which wasp eggs hatch, [6, 9]. The way through which the bacterial infection occurs within the aphid is not yet fully understood, altough it has been documented to occur sporadically, [3]. Such a situation has been modeled in [4], focussing on aphids that “may harbour the facultative bacterial endosymbionts”, with hosts that become infected through vectors and may also get rid of their parasites. A mathematical model proposing a different transmission mechanism for the bacterial infection, through the wasp’s oviposition, has instead been studied in [8].

Other aphids antagonists are generally represented by predators and pathogens. This gives alternative biological ways of controlling the aphids populations. In this setting, pathogenic fungi can infect the insects with which their spores come in contact and potentially represent another possible another different way of keeping these pests in check.

In this paper, we propose a mathematical model to investigate the three-way interactions between the crops, that is the human resource, which is however hidden in the model, not explicitly taken into account as a dependent variable, aphids and fungi.

Aphids, along with whiteflies and scale insects, are a specialised group of herbivorous insects. Their mouth parts form a stylet, which they use to pierce plant tissues and thus gain direct access to the plant phloem and its constant stream of nutrients. This feeding habit links the insect to the plant as if the insects were just another plant organ such as a leaf. In effect, the insect population acts as an extra sink, which diverts carbon and nitrogen from the plant proper. Plant nutrients flow cleanly into the insects without the risk of accidental ingestion of pathogens. However, fungal entomopathogens have evolved to invade these sap-sucking insects not through the guts but through penetration of the insect cuticle. Infected insects upon dying turn into fungal spore-producing cadavers. Spores are actively projected from the cadaver and can infect nearby insects. A race between insect host and pathogen ensues, governed by the respective reproductive rates of the insect and the pathogen, as well as by the pathogen’s ability to spread and infect. Some spores are specialised for diapause and form resting spores, which can remain dormant in the environment for several years [1]. In this study, we focus on aphids attacking cereals in temperate regions of the world. In such conditions, aphids have a complex life cycle shifting between winter and summer plant hosts. In the spring, aphids leave their winter host and settle in the cereal fields. Depending on weather conditions, e.g. temperature, rainfall, sunlight and the presence of aphid antagonists, such as predators, parasitoids and pathogens, aphid populations may increase extremely fast and reach crop-damaging densities. In the field, aphids reproduce parthenogenetically, giving rise to unwinged offsprings that stay in the field. As the crop ripens and the phloem dries out, an increasing proportion of the offsprings will develop wings by which they are able to leave the field. High densities of aphids as well will also induce wing formation enabling the aphids to escape intraspecific competition, [2]. We consider the tri-trophic system consisting of cereal-aphids-fungus in the time period from spring until harvest. In natural conditions, while the fungus is always present, aphids instead immigrate from the surroundings areas in the cereal field. Aphids are lost from the system, because they are killed by natural enemies or by infection by the fungus, or by winged emigration.

Let

The model is summarised by the compartmental diagram in Figure 1.

The plant biomass

For most purposes it is easier to work with the equivalent (

In this section we want to ensure that the system (1) does not have unbounded solutions, for (realistic) initial conditions in a suitable set, because otherwise the trajectories would not have biological significance.

We seek a positively invariant set

for some constant

On the part of the boundary where

On the part of the boundary where

which holds if and only if

So for positive invariance we need

Finally, on the part of the boundary where

So we have a positively invariant set as long as we choose

which we can do.

We shall start by analysing the steady states of the system, the solutions of _{0}, 0, 0). This automatically satisfies

giving the non-trivial solution

This is biologically realistic (positive) if

or in other words if the basic growth rate for aphids is greater than their basic death rate. It is easy to show that if (4) does not hold, the solution of the system of differential equations tends to (0, 0, 0), so the aphids and the fungus die out, as would be expected. We shall consider from now on the case that (4) is satisfied.

We now seek non-trivial (enzootic) steady states of the form (^{*}, ^{*}, ^{*}), with none of ^{*}, ^{*} and ^{*} equal to zero. The ^{*} = ^{*}, which with the

Since ^{*} ≠ 0, then

and so

From the

Eliminating ^{*} between equations (6) and (7), we obtain

or

a quadratic equation for ^{*} with two real roots, one positive and one negative. The negative root clearly does not give a biologically realistic steady state, and we let ^{*} denote the positive root from now on. In order to have a biologically meaningful steady state we observe that for ^{*} and from (5) for ^{*} = ^{*} − ^{*}, the values are nonnegative, but we have to check whether (6) also gives a positive value. However, observe that

so ^{*} > 0 as long as ^{*} < _{0}, or equivalently (see diagram) as long as _{0}) > 0, or, given the value of _{0},

This is never true (for realistic _{0}) if _{0} > _{0} <

Equivalently, in view of (3), it is true for

There is a transcritical bifurcation as ^{*}, ^{*}, ^{*}) enters the positive octant by passing through the semi-trivial steady state (_{0}, 0, 0) when it is at (^{*}, ^{*}, ^{*}) becomes stable and (_{0}, 0, 0) unstable as ^{*} < _{0} for ^{*} increases with

and the formula for the solution of a quadratic shows that the denominator is positive. It follows from this that the fraction ^{*}/^{*} of infected aphids decreases with

The bifurcation diagram with the bifurcations we have seen so far is sketched in Figure 4. There is no further bifurcation simply involving steady states, but Hopf bifurcations have not yet been ruled out.

The Jacobian matrix

At (0, 0, 0), _{0}, 0, 0), _{0} = _{0}, 0, 0) has eigenvalues _{0} = − (_{0} are given by

This matrix has negative trace, and so has stable eigenvalues as long as its determinant is positive, that is as long as

This is always true if _{0}, 0, 0) is stable whenever ^{*}, ^{*}, ^{*}) exists and is realistic. Its stability is determined by the eigenvalues of ^{*} = ^{*}, ^{*}, ^{*}), which can be simplified slightly to

where

and ^{*} = ^{*} − ^{*}. The eigenvalues ^{*} satisfy its characteristic equation

where

and

The (Routh--Hurwitz) conditions for stability of (^{*}, ^{*}, ^{*}) are that _{1} > 0, _{3} > 0, and _{1}_{2}−_{3} > 0. At a Hopf bifurcation, _{1}_{2}−_{3} = 0.

We have not analysed the possibility of Hopf bifurcations in the general case, but an analysis with _{0}, 0, 0) enters the positive octant and becomes stable and another close to it at ^{*}, ^{*}, ^{*}) does. We now consider what happens as ^{*}, ^{*}, ^{*}) is stable, for

and

so that

All of these are positive and ^{*}, ^{*}, ^{*}) is that _{1}_{2} − _{3} > 0, which is satisfied since _{1}_{2} = ^{2}) while _{3} = ^{*}, ^{*}, ^{*}) is stable whenever

So (^{*}, ^{*}, ^{*}) is stable when it enters the positive quadrant at _{1}, _{1} and _{1} by _{1}, _{1}, and _{1} = _{1}, where

Looking for solutions

and

Definition (13) gives

With _{0} + _{1},

From (15) and (16), the coefficients _{1}, _{2} and _{3} of the characteristic equation (14) for ^{*} are given to leading order by

and

Both _{1} and _{3} are positive as long as _{1} > 0. By Routh–Hurwitz conditions, the steady state is stable as long as _{1}_{2} − _{3} > 0, or equivalently as long as _{1}_{21} − _{31} > 0. However,

where

Hence _{1}_{2} − _{3} is positive, and so (^{*}, ^{*}, ^{*}) is stable, for _{1} small and positive, as expected. It increases as _{1} increases if ^{*}, ^{*}, ^{*}) remains stable at least in this asymptotic regime. It decreases as _{1} increases if _{1} = _{1}, where

Using the expression (10) for

Under this condition, as _{1} increases past _{1}, or _{1}, (^{*}, ^{*}, ^{*}) loses stability through a Hopf bifurcation. We have shown that it regains stability later, and that this must be through a reverse Hopf bifurcation before

_{0}

The basic reproduction number _{0} can be thought of as the number of infected aphids produced in each generation of the infection when an infected aphid is introduced into the disease-free system, which is the system at (_{0}, 0, 0). Indeed, note that in this case, an infected aphid produces fungal particles in the next infection generation, which produce infected aphids in the infection generation after that; hence, we shall consider the number of infected aphids produced over two infection generations. We assume implicitly that (_{0}, 0, 0) exists and is realistic, in other words that (4) holds. _{0} may be shown to be the largest eigenvalue of the next-generation matrix _{0}, 0, 0), which is given after a standard calculation by

This matrix has eigenvalues

Note that _{0} < 1, since

If _{+} < 1, then all eigenvalues of _{0} < 1, the disease cannot invade, and the disease-free steady state is stable. On the other hand if _{+} > 1, then _{0} = _{+} > 1, and the fungus invades the disease-free steady states.

In the case of interest,

This may be interpreted as follows, with arguments that can be made mathematically rigorous. An infected aphid introduced into the system at (_{0}, 0, 0), the primary, leaves the infected aphid (_{0} and so remains in the _{0}), on average. While it is in the class it produces fungus _{0}) fungal particles. Each fungal particle leaves the _{0}, so it produces on average _{0}/_{0} per generation.

The condition for invasion, _{0} > 1, may be written as

or, as we saw in section 5, _{0}) > 0, which is equivalent to ^{*} < _{0}, and then to ^{*} > 0 as well, which is identical to the condition for (^{*}, ^{*}, ^{*}) to have all components positive. Thus, whenever the diseased steady state (^{*}, ^{*}, ^{*}) is realistic, then the disease-free steady state (_{0}, 0, 0) is unstable to the introduction of the disease.

Using XPPAUT and Matlab, we performed some numerical simulations that are here reported. In Figure 4, the one parameter bifurcation diagram of system (1) is represented. The populations

In the panels of Figure 7, the possible combinations of the parameter

The three-way interaction ecosystem formed by crops, aphids and fungi has been shown to possess only three possible equilibria. Namely, ecosystem collapse, the disease-free state and the enzootic coexistence point. Furthermore, coexistence may occur also through persistent oscillations, demonstrated by the results of the numerical simulations.

More specifically, the system dies out unless the parameter _{0}, 0, 0) unless the transmission parameters are sufficiently high (^{*}, ^{*}, ^{*}) is stable for all values of

The role of the parameter ^{*}; it also favours the pathogen by increasing ^{*}, because of course the host is an essential resource for the pathogen. However, from (11) although high, ^{*}/N^{*} of infected hosts. The host cannot escape the pathogen by growing quickly, but it can reduce its effect on the host population, so that biological control of a fast-growing pest will be difficult.

More generally, the biocontrol of crops in this situation should aim at obtaining the coexistence state and avoiding the disease-free point, as the infection is damaging the crop pests, and therefore it is useful to preserve and improve the harvesting. In the case of persistent oscillations on the other hand, the troughs should possibly be kept at a high level, to prevent the population of pathogenic fungi from disappearing.

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