Just a simple idea after observation, penny worth. And this is what I could say, no more.
You have only 10 data points. For a model among models, you might need strong assumptions about the data.
Time series? How strongly you believe the data be a time series, or you could think they were i.i.d.
Prior knowledge? Do you have any knowledge about meat and eggs production and consumption in that province and also China?
Related facts? Do you have any other data that correlated with your 10 point data? (So you have more than just  these 10 points)
Empirical analysis instead of theoretical application. It seems that classical statistics would not solve your problem. Bayesian method could be better as it puts your prior knowledge and assumptions in action.

尝试一下非经典方法吧，统计讲究的是从样本推断总体，然而你的样本数过少，怎么能代表总体呢？就比如你要研究中国人平均收入状况你怎么能在样本数极少而且没用通过任何的抽样技术下去推断呢？如果楼主想纯粹的从模型角度出发，我想你可以尝试下黄崇福的信息扩散理论或者邓聚龙的灰色理论，虽然我很鄙视这种类似于“老子生儿子然后儿子再生老子”等在哲学上行不通的方法，但还是建议你去做做。这些方法论的诞生都是在中国特色国情--数据不能共享的情况下下产生的，反正你也无法检验，是吧！别鄙视我的建议，方法论而已！

邓聚龙的灰色理论