Skip to main content
Log in

Mate Finding, Sexual Spore Production, and the Spread of Fungal Plant Parasites

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

Sexual reproduction and dispersal are often coupled in organisms mixing sexual and asexual reproduction, such as fungi. The aim of this study is to evaluate the impact of mate limitation on the spreading speed of fungal plant parasites. Starting from a simple model with two coupled partial differential equations, we take advantage of the fact that we are interested in the dynamics over large spatial and temporal scales to reduce the model to a single equation. We obtain a simple expression for speed of spread, accounting for both sexual and asexual reproduction. Taking Black Sigatoka disease of banana plants as a case study, the model prediction is in close agreement with the actual spreading speed (100 km per year), whereas a similar model without mate limitation predicts a wave speed one order of magnitude greater. We discuss the implications of these results to control parasites in which sexual reproduction and dispersal are intrinsically coupled.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Allee WC, Park O, Emerson AE, Park T, Schmidt KP et al (1949) Principles of animal ecology, 1st edn. WB Saunders Co., Ltd, Philadelphia

    Google Scholar 

  • Alsos IG, Eidesen PB, Ehrich D, Skrede I, Westergaard K, Jacobsen GH, Landvik JY, Taberlet P, Brochmann C (2007) Frequent long-distance plant colonization in the changing arctic. Science 316(5831):1606–1609

    Article  Google Scholar 

  • Aylor DE (2003) Spread of plant disease on a continental scale: role of aerial dispersal of pathogens. Ecology 84(8):1989–1997

    Article  Google Scholar 

  • Baker HG (1955) Self-compatibility and establishment after ‘long-distance’ dispersal. Evolution 9(3):347–349

    Article  Google Scholar 

  • Billiard S, López-Villavicencio M, Devier B, Hood ME, Fairhead C, Giraud T (2011) Having sex, yes, but with whom? inferences from fungi on the evolution of anisogamy and mating types. Biol Rev 86(2):421–442

    Article  Google Scholar 

  • Bonner JT (1958) The relation of spore formation to recombination. Am Nat 92:193–200

    Article  Google Scholar 

  • Brown JK, Hovmøller MS (2002) Aerial dispersal of pathogens on the global and continental scales and its impact on plant disease. Science 297(5581):537–541

    Article  Google Scholar 

  • Bultman TL, White JF Jr, Bowdish TI, Welch AM, Johnston J (1995) Mutualistic transfer of Epichloë spermatia by Phorbia flies. Mycologia 87(2):182–189

    Article  Google Scholar 

  • Burie JB, Calonnec A, Ducrot A (2006) Singular perturbation analysis of travelling waves for a model in phytopathology. Math Model Nat Phenom 1(1):49–62

    Article  MathSciNet  MATH  Google Scholar 

  • Burie JB, Calonnec A, Langlais M (2008) Modeling of the invasion of a fungal disease over a vineyard. In: Mathematical modeling of biological systems, volume II, Springer, pp 11–21, http://www6.bordeaux-aquitaine.inra.fr/sante-agroecologie-aquitaine.inra.fr/sante-agroecologie-vignoble/layout/set/print/content/download/3769/36021/file/Pub13-ACL-Burie-07.pdf

  • Castel M, Mailleret L, Andrivon D, Ravigné V, Hamelin FM (2014) Allee effects and the evolution of polymorphism in cyclic parthenogens. Am Nat 183(3):E75–E88

    Article  Google Scholar 

  • Cheptou PO (2012) Clarifying baker’s law. Ann Bot 109(3):633–641

    Article  Google Scholar 

  • Cheptou PO, Massol F (2009) Pollination fluctuations drive evolutionary syndromes linking dispersal and mating system. Am Nat 174(1):46–55

    Article  Google Scholar 

  • Cunniffe NJ, Gilligan CA (2008) Scaling from mycelial growth to infection dynamics: a reaction diffusion approach. Fungal Ecol 1(4):133–142

    Article  Google Scholar 

  • Desprez-Loustau ML, Robin C, Buée M, Courtecuisse R, Garbaye J, Suffert F, Sache I, Rizzo DM (2007) The fungal dimension of biological invasions. Trends Ecol Evol 22(9):472–480

    Article  Google Scholar 

  • Ene IV, Bennett RJ (2014) The cryptic sexual strategies of human fungal pathogens. Nat Rev Microbiol 12(4):239–251

    Article  Google Scholar 

  • Fisher MC, Henk DA, Briggs CJ, Brownstein JS, Madoff LC, McCraw SL, Gurr SJ (2012) Emerging fungal threats to animal, plant and ecosystem health. Nature 484(7393):186–194

    Article  Google Scholar 

  • Fouré E (1982) Les cercosporioses du bananier et leurs traitements. comportement des variétés. 1: Incubation et évolution de la maladie. Fruits 37(12):749–766

    Google Scholar 

  • Frantzen J, van den Bosch F (2000) Spread of organisms: can travelling and dispersive waves be distinguished? Basic Appl Ecol 1(1):83–92

    Article  Google Scholar 

  • Garrett K, Bowden R (2002) An Allee effect reduces the invasive potential of Tilletia indica. Phytopathology 92(11):1152–1159

    Article  Google Scholar 

  • Gascoigne J, Berec L, Gregory S, Courchamp F (2009) Dangerously few liaisons: a review of mate-finding allee effects. Pop Ecol 51(3):355–372

    Article  Google Scholar 

  • Gross A, Holdenrieder O, Pautasso M, Queloz V, Sieber TN (2014) Hymenoscyphus pseudoalbidus, the causal agent of European ash dieback. Mol Plant Pathol 15(1):5–21

    Article  Google Scholar 

  • Gurney W, Nisbet R (1975) The regulation of inhomogeneous populations. J Theor Biol 52(2):441–457

    Article  MathSciNet  Google Scholar 

  • Halkett F, Coste D, Rivas Platero GG, Zapater MF, Abadie C, Carlier J (2010) Genetic discontinuities and disequilibria in recently established populations of the plant pathogenic fungus Mycosphaerella fijiensis. Mol Ecol 19(18):3909–3923

    Article  Google Scholar 

  • Husson C, Scala B, Caël O, Frey P, Feau N, Ioos R, Marçais B (2011) Chalara fraxinea is an invasive pathogen in France. Eur J Plant Pathol 130(3):311–324

    Article  Google Scholar 

  • Landry C (2015) Modélisation des dynamiques de maladies foliaires de cultures pérennes tropicales différentes échelles spatiales: cas de la cercosporiose noire du bananier. PhD thesis, Université des Antilles, http://www.theses.fr/2015AGUY0835

  • Lewis M, Kareiva P (1993) Allee dynamics and the spread of invading organisms. Theor Pop Biol 43(2):141–158

    Article  MATH  Google Scholar 

  • Lutscher F (2008) Density-dependent dispersal in integrodifference equations. J Math Biol 56(4):499–524

    Article  MathSciNet  MATH  Google Scholar 

  • Mallet J (2001) Insect movement: mechanisms and consequences. In: Woiwod I, Reynolds D, Thomas C (eds) Gene flow. CABI, pp 337–360

  • Massol F, Cheptou P (2011a) When should we expect the evolutionary association of self-fertilization and dispersal? Evolution 65(5):1217

    Article  Google Scholar 

  • Massol F, Cheptou PO (2011b) Evolutionary syndromes linking dispersal and mating system: the effect of autocorrelation in pollination conditions. Evolution 65(2):591–598

    Article  Google Scholar 

  • Medlock J, Kot M (2003) Spreading disease: integro-differential equations old and new. Math Biosci 184(2):201–222

    Article  MathSciNet  MATH  Google Scholar 

  • Morel-Journel T, Girod P, Mailleret L, Auguste A, Blin A, Vercken E (2015) The highs and lows of dispersal: how connectivity and initial population size jointly shape establishment dynamics in discrete landscapes. Oikos. doi:10.1111/oik.02718

  • Mundt CC, Sackett KE, Wallace LD, Cowger C, Dudley JP (2009) Long-distance dispersal and accelerating waves of disease: empirical relationships. Am Nat 173(4):456–466

    Article  Google Scholar 

  • Murray JD (2002) Mathematical biology I: an introduction, interdisciplinary applied mathematics, 3rd edn. Springer, New York

    Google Scholar 

  • Pannell JR, Auld JR, Brandvain Y, Burd M, Busch JW, Cheptou PO, Conner JK, Goldberg EE, Grant AG, Grossenbacher DL et al (2015) The scope of baker’s law. New Phytol 208:656–667

    Article  Google Scholar 

  • Pernaci M, De Mita S, Andrieux A, Pétrowski J, Halkett F, Duplessis S, Frey P (2014) Genome-wide patterns of segregation and linkage disequilibrium: the construction of a linkage genetic map of the poplar rust fungus Melampsora larici-populina. Front Plant Sci 5:712

    Article  Google Scholar 

  • Powell JA, Slapničar I, van der Werf W (2005) Epidemic spread of a lesion-forming plant pathogen-analysis of a mechanistic model with infinite age structure. Linear Algebra Appl 398:117–140

    Article  MathSciNet  MATH  Google Scholar 

  • Rieux A, Lenormand T, Carlier J, Lapeyre de Bellaire L, Ravigné V (2013) Using neutral cline decay to estimate contemporary dispersal: a generic tool and its application to a major crop pathogen. Ecol Lett 16(6):721–730

    Article  Google Scholar 

  • Rieux A, Soubeyrand S, Bonnot F, Klein EK, Ngando JE, Mehl A, Ravigne V, de Bellaire LdL (2014) Long-distance wind-dispersal of spores in a fungal plant pathogen: estimation of anisotropic dispersal kernels from an extensive field experiment. PloS ONE 9(8):e103,225

    Article  Google Scholar 

  • Robert S (2012) Emergence mondiale de la maladie des raies noires du bananier: histoire de l’invasion et stratégie de vie du champignon phytopathogène Mycosphaerella fijiensis. PhD thesis, Université de Montpellier 2

  • Roques L, Garnier J, Hamel F, Klein EK (2012) Allee effect promotes diversity in traveling waves of colonization. Proc Natl Acad Sci USA 109(23):8828–8833

    Article  MathSciNet  Google Scholar 

  • Roy B (1994) The effects of pathogen-induced pseudoflowers and buttercups on each other’s insect visitation. Ecology 75(2):352–358

    Article  Google Scholar 

  • Shaw A, Kokko H (2015) Dispersal evolution in the presence of allee effects can speed up or slow down invasions. Am Nat 185(5):631

    Article  Google Scholar 

  • Soubeyrand S, Laine AL, Hanski I, Penttinen A (2009) Spatiotemporal structure of host-pathogen interactions in a metapopulation. Am Nat 174(3):308–320

    Article  Google Scholar 

  • Stokes A (1976) On two types of moving front in quasilinear diffusion. Math Biosci 31(3):307–315

    Article  MathSciNet  MATH  Google Scholar 

  • Stover R (1980) Sigatoka leaf spots of bananas. Plant Dis 64(8):751

    Article  Google Scholar 

  • Taylor CM, Hastings A (2005) Allee effects in biological invasions. Ecol Lett 8(8):895–908

    Article  Google Scholar 

  • Travis JM, Murrell DJ, Dytham C (1999) The evolution of density-dependent dispersal. Proc R Soc B 266(1431):1837–1842

    Article  Google Scholar 

  • Veit RR, Lewis MA (1996) Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion of eastern North America. Am Nat 148:255–274

    Article  Google Scholar 

  • Vercken E, Kramer A, Tobin P, Drake J (2011) Critical patch size generated by allee effect in gypsy moth, Lymantria dispar (L.). Ecol Lett 14(2):179–186

    Article  Google Scholar 

  • Williams GC (1975) Sex and evolution. Princeton University Press, Princeton

    Google Scholar 

  • Wilson W, Harder L (2003) Reproductive uncertainty and the relative competitiveness of simultaneous hermaphroditism versus dioecy. Am Nat 162(2):220–241

    Article  Google Scholar 

Download references

Acknowledgments

MAL gratefully acknowledges a Canada Research Chair, a Killam Fellowship, and NSERC Discovery and Accelerator awards. VR benefited from funds from project BIOFIS (reference 1001-001) of the Agropolis Fondation (Montpellier, France). FMH acknowledges funding from the French National Research Agency (ANR) as part of the ‘Blanc 2013’ programme (ANR-13-BSV7-0011, FunFit project) and from the French National Institute for Agricultural Research (INRA) ‘Plant Health and the Environment’ Division. FMH thanks Thomas Hillen for early mathematical feedback and Isaline Aubert and Antoine Ollivier for their contributions to this work through short internship periods. We thank the two anonymous reviewers for their helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Frédéric M. Hamelin.

Appendix: Analogous Analysis Without Mate Limitation

Appendix: Analogous Analysis Without Mate Limitation

In this section, sexual spores can be produced regardless of the presence of a mate, which amounts to removing the probability (i / 2) / n in Eq. (1). Without mate limitation, model (1) becomes

$$\begin{aligned} u_t= & {} \sigma i-\mu u+\kappa u_{xx}\,,\nonumber \\ i_t= & {} \left( q\mu u+p\alpha i\right) (n-i)/n\,. \end{aligned}$$
(15)

We rescale variables according to

$$\begin{aligned} t^*=\frac{t}{T}\,,\quad x^*=\frac{x}{L}\,,\quad i^*=\frac{i}{n}\,,\quad u^*=\frac{\mu }{\sigma n}\times u\,, \end{aligned}$$

and

$$\begin{aligned} a=\frac{1}{q\sigma T}\,,\quad \varepsilon =\frac{1}{\mu T}\,, \quad b=\frac{p\alpha }{q\sigma }\,,\quad d=\sqrt{\frac{\kappa }{\mu L^2}}\,. \end{aligned}$$

Dropping the asterisks for convenience, model (15) reads:

$$\begin{aligned} ai_t= & {} (u+ b i)(1-i)\,,\nonumber \\ \varepsilon u_t= & {} i - u+ d^2 u_{xx}\,. \end{aligned}$$
(16)

Applying the quasi-steady-state approximation to the second equation of (16) yields the following nonlocal integrodifferential equation for i:

$$\begin{aligned} ai_t=(1-i)\left( b i+\int _{-\infty }^{\infty } H(x-y)i(y,t)\,\mathrm {d}y\right) \,, \end{aligned}$$
(17)

where H is the Laplace kernel (6).

A Taylor expansion of i(yt) at x in (17) yields

$$\begin{aligned} \int _{-\infty }^{\infty } H(x-y)i(y,t)\,\mathrm {d}y= & {} i(x,t)\int _{-\infty }^{\infty } H(x-y)\,\mathrm {d}y\nonumber \\&\quad +\, i_x(x,t)\int _{-\infty }^{\infty } (x-y)H(x-y)\,\mathrm {d}y\nonumber \\&\quad + \,\frac{i_{xx}(x,t)}{2}\int _{-\infty }^{\infty } (x-y)^2H(x-y)\,\mathrm {d}y\nonumber \\&\quad + \,\cdots \,. \end{aligned}$$
(18)

Using the moments of the Laplace distribution in (18), we get

$$\begin{aligned} ai_t = (1-i)(bi+i+d^2i_{xx}+d^4i_{xxxx}+\ldots )\,. \end{aligned}$$
(19)

A second-order Taylor expansion as d goes to zero in (19) yields

$$\begin{aligned} a i_t\approx (1-i)\left( b i+i+d^2i_{xx}\right) \,. \end{aligned}$$
(20)

Equation (20) can be expressed in a travelling wave form as

$$\begin{aligned} -\hat{c} I'=(1-I)\left( \left( b+1\right) I+I''\right) \,, \end{aligned}$$
(21)

where \(\hat{c}=ac/d\), and the prime denotes differentiation w.r.t. \(z=(x-ct)/d\). Proceeding as in Sect. 3.2, Eq. (21) can be expressed as a dynamical system:

$$\begin{aligned} \dot{I}= & {} J(1-I)\,,\nonumber \\ \dot{J}= & {} -{\hat{c}} J-(1-I)(b+1)I\,. \end{aligned}$$
(22)

Its equilibria are \((I,J)=(0,0)\) and (1, 0). Let G be the associated Jacobian matrix:

$$\begin{aligned} G=\left( \begin{array}{cc}-J&{}1-I\\ -(b+1)(1-2I)&{}-{\hat{c}}\end{array}\right) \,. \end{aligned}$$

Linearizing around (0, 0), we get

$$\begin{aligned} G_{(0,0)}=\left( \begin{array}{cc} 0&{}1\\ -(b+1)&{}-{\hat{c}}\end{array}\right) \,, \end{aligned}$$

whose eigenvalues are

$$\begin{aligned} \lambda _\pm =\frac{-{\hat{c}}\pm \sqrt{{\hat{c}}^2-4(b+1)}}{2}\,. \end{aligned}$$

So (0, 0) is a stable node if \({\hat{c}}>2\sqrt{b+1}\), and a stable spiral otherwise. From Murray (2002)’s Sect. 13.2 (Eq. 13.12 and Fig. 13.1), we conjecture that there is a trajectory from (1, 0) to (0, 0) lying entirely in the quadrant \(I\ge 0\), \(J\le 0\) with \(0\le I\le 1\) for all wave speeds \({\hat{c}}\ge {\hat{c}}^\star \), with

$$\begin{aligned} {\hat{c}}^\star =2\sqrt{b+1}\,. \end{aligned}$$

We numerically checked that only \({\hat{c}}^\star \) is relevant, greater wave speeds being unstable. In terms of the original dimensional Eq. (15), the wave speed can be expressed as

$$\begin{aligned} c=q\sigma \sqrt{\frac{\kappa }{\mu }}2\sqrt{\frac{p\alpha }{q\sigma }+1}\,. \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hamelin, F.M., Castella, F., Doli, V. et al. Mate Finding, Sexual Spore Production, and the Spread of Fungal Plant Parasites. Bull Math Biol 78, 695–712 (2016). https://doi.org/10.1007/s11538-016-0157-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-016-0157-1

Keywords

Mathematics Subject Classification

Navigation