A mathematical model on the effect of M. denticulata weed on different winter crops

Many weeds interfere with the productivity of wheat and other winter annual crops; however, one common weed Medicago denticulata, appears to have a beneficial effect. An experiment was conducted at Agricultural Experimental Farm of the Indian Statistical Institute, Giridih, Jharkhand to observe the effect of M. denticulata (a legume) on different crops yield of barley (Hordeum vulgare Sensu lato), linseed (Linum usitatissimum L.), Indian mustard [Brassica juncea (L.) Czernj. & Cosson] and wheat (Triticum aestivum L.). The commensalisms interaction was studied, in which one species benefits, while the other is unaffected, between the crops, M. denticulata and other weeds, to observe the role of medicago as biological control agent of other weeds for enhancement of crop productivity. Further, a statistical test was performed to establish the effect of M. denticulata and weed infestations based on one-step Markov structure of the covariance matrix of the biomass. This permitted development of a mathematical model of a crop–M. denticulata–other weeds system and their interactions through direct and indirect competition for resources. Crops yield decreased considerably due to the presence of other weeds, while it increased due to introduction of medicago and hence medicago acted as a controlling agent in a crop–M. denticulata–other weeds system. Our mathematical analysis supported the experimental findings. The model parameters are estimated through a nonlinear least square and stochastic differential equations. Moreover, some thresholds for the growth of medicago population came out from our analysis that may be used as a biological control threshold for the enhancement of crop productivity.